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COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK 

PUBLICATION NUMBER TWO OF THE 
ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH 

ESTABLISHED DBOBMBEB 17tli 1904 



THE THEORY OF ELECTRONS 

AND ITS APPLICATIONS TO THE PHENOMENA 
OF LIGHT AND RADIANT HEAT 



A COURSE OF LECTURES DELIVERED IN COLUMBLi 
UNIVERSITY, NEW YORK, IN MARCH AND APRIL 1906 

BY 

H. A. LORENTZ 

PEOFESSOB IN THE UNIVEBSITY OF LEIDEN 
LECTUKKB IN MATHEMATICAI. PHYSIOS IN OOtUMBIA UNIVBKBITY FOB 1905-1906 



NEW YOEK 

THE COLUMBIA UNIVEESITY PKESS 

THE MACMILLAN COMPANY, Agents 

LONDON: MACMILLAN CO., Ltd. 
1909 



COLUMBIA UNIVERSITY IN THE CITY OP NEW YORK 

PUBLICATION NUMBER TWO OF THE 

ERNEST KEMPTON ADA.M8 FUND FOR PHYSICAL RESEARCH 

ESTABLISHED DECEMBER 17th, 1904 



THE THEOKY OF ELECTRONS 

AND ITS APPLICATIONS TO THE PHENOMENA 
OF LIGHT AND EADIANT HEAT 



A COURSE OF LECTURES DELIVERED IN COLUMBIA 
UNIVERSITY, NEW YORK, IN MARCH AND APRIL, 1906 



BY 



H. A. LORENTZ 



PROFESSOR IN THE UNIVERSITY OF LEIDEN 
LECTUKER IN MA.TIIEUATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1905-1906 



NEW YORK 

THE COLUMBIA UNIVERSITY^ PEESS 

THE MACMILLAN CO., Agents 

LONDON: MACMILLAN & CO., Ltd. 

1909 



Oq the seventeenth day of December, nineteen hundred and 
four, Edward Dean Adams, of New York, established in Cohimbia 
University "The Ernest Kempton Adams Fund for Physical 
Research" as a memorial to his son, Ernest Kempton Adams, 
who received the degrees of Electrical Engineering in 1897 and 
Master of Arts in 1898, and who devoted his life to scientific re- 
search. The income of this fund is, by the terms of the deed of 
gift, to be devoted to the maintenance of a research fellowship and 
to the publication and distribution of the results of scientific re- 
search on the part of the fellow. A generous interpretation of 
the terms of the deed on the part of Mr. Adams and of the Trus- 
tees of the University has made it possible to issue and distribute 
a limited number of copies of the following lectures as a publi- 
cation of the Ernest Kempton Adams Fund : 



B. G. TEUBNERS SAMMLUNG VON LEHRBUCHERN 

AUF DEM GEBIETB DER 

MATHEMATISCHEN WISSENSCHAFTEN 

MIT BINSCHLUSS IHRER ANWENDUNGEN. 
BAND XXIX. 



THE THEORY OE ELECTRONS 

AND ITS APPLICATIONS TO THE PHENOMENA 
OF LIGHT AND EADIANT HEAT 



A COURSE OF LECTURES DELIVERED IN COLUMBIA 
UNIVERSITY, NEW YORK, IN MARCH AND APRIL 1906 

BY 

H. A. LpRENTZ 

PEOFESSOK IN THE UNIVERSITY OF LEIDEN 
LECTURER IN MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1905-1906 



LEIPZIG: B. G. TEUBNER 

1909 

LONDON : DAVID NUTT, 57-59 long acre - WILLIAMS & NORGATE, w. c. u Henrietta street 
NEW YORK; G. E, STECHERT & Co., 129-133 west 20™ street 

T) 



AIL EIGHTS EBSBEVED. 
PRINTED BY E.G.TBUBNBE, LEIPZIG. 



PREFACE. 

The publication of these lectures, which I delivered in Columbia 
University in the spring of 1906, has been unduly delayed, chiefly 
on account of my wish to give some further development to the sub- 
ject, so as to present it in a connected and fairly complete form; 
for this reason I have not refrained from making numerous additions. 
Nevertheless there are several highly interesting questions, more or 
less belonging to the theory of electrons, which I could but slightly 
touch upon. I could no more than allude in a note to Voigt's 
Treatise on magneto-optical phenomena, and neither Planck's views 
on radiation, nor Einstein's principle of relativity have received an 
adequate treatment. 

In one other respect this book wiU, I fear, be found very deficient. 
No space could be spared for a discussion of the different ways in 
which the fundamental principles may be established, so that, for in- 
stance, there was no opportunity to mention the important share that 
has been taken in the development of the theory by Larmor and 
Wiechert. 

It is with great pleasure that I express my thanks to Professor 
A. P. Wills for his kindness in reading part of the proofs, and to 
the publisher for the care he has bestowed on my work. 

Leiden, January 1909. 

H. A. Lorentz. 



CONTENTS. 

Chapter 

I. General principles. Theory of free electrons. 1 

11. Emission and absorption of heat . . . . 68 

m. Theory of the Zeeman-effect 98 

IV. Propagation of light in a body composed of molecules. Theory of the 

inverse Zeeman-eflFect 132 

V. Optical phenomena in moving bodies . . 168 

Notes . .231 

Index 330 



CHAPTER I. 

GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 

Tlie theory of electrons, on which I shall have the honor to 
lecture before you, already forms so vast a subject, that it will be 
impossible for me to treat it quite completely. Even if I confine 
myself to a general review of this youngest branch of the science 
of electricity, to its more important applications in the domain 
of light and radiant heat, and to the discussion of some of the 
difficulties that still remain, I shall have to express myself as con- 
cisely as possible, and to use to the best advantage the time at our 
disposal. 

In this, as in every other chapter of mathematical physics, we 
may distinguish on the one hand the general ideas and hypotheses 
of a physical nature involved, and on the other the array of 
mathematical formulae and developments by which these ideas and 
hypotheses are expressed and worked out. I shall try to throw a 
clear light on the former part of the subject, leaving the latter part 
somewhat in the background and omitting all lengthy calculations, 
which indeed may better be presented in a book than in a lecture.^) 

1. As to its physical basis, the theory of electrons is an off- 
spring of the great theory of electricity to which the names of 
Faraday and Maxwell wiU. be for ever attached. 

You all know this theory of Maxwell, which we may call the 
general theory of the electromagnetic field, and in which we con- 
stantly have in view the state of the matter or the medium by which 
the field is occupied. While speaking of this state, I must immediately 
call your attention to the curious fact that, although we never lose 
sight of it, we need by no means go far in attempting to form an 
image of it and, in fact, we cannot say much about it. It is true 
that we may represent to ourselves internal stresses existing in the 



1) In this volume sucli calculations as I have only briefly indicated in my 
lectures are given at full length in the appendix at the end. 

Lorentz, Theory of electrons. 1 



2 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

medium surrounding an electrified body or a magnet, that we may 
think of electricity as of some substance or fluid, free to move in 
a conductor and bound to positions of equilibrium in a dielectric, 
and that we may also conceive a magnetic field as the seat of 
certain invisible motions, rotations for example around the lines of 
force. All this has been done by many physicists and Maxwell 
himself has set the example. Yet, it must not be considered as 
really necessary; we can develop the theory to a large extent and 
elucidate a great number of phenomena, without entering upon 
speculations of this kind. Indeed, on account of the difficulties into 
which they lead us, there has of late years been a tendency to avoid 
them altogether and to establish the theory on a few assumptions 
of a more general nature. 

The first of these is, that in an electric field there is a certain 
state of things which gives rise to a force acting on an electrified 
body and which may therefore be symbolically represented by the 
force acting on such a body per unit of charge. This is what we 
call the electric force, the symbol for a state in the medium about 
whose nature we shall not venture any further statement. The second 
assumption relates to a magnetic field. Without thinking of those 
hidden rotations of which I have just spoken, we can define this by 
the so called magnetic force, i. e. the force acting on a pole of unit 
strength. 

After having introduced these two fundamental quantities, we 
try to express their mutual connexions by a set of equations which 
are then to be applied to a large variety of phenomena. The mathe- 
matical relations have thus come to take a very prominent place, 
so that Hertz even went so far as to say that, after all, the theory 
of Maxwell is best defined as the system of Maxwell's equations. 

We shall not use these formulae in the rather complicated form 
in which they can be found in Maxwell's treatise, but in the clearer 
and more condensed form that has been given them by Heaviside 
and Hertz. In order to simplify matters as much as possible, I shall 
further introduce units ^) of such a kind that we get rid of the larger 
part of such factors as 43r and 1/4 ;w, by which the formulae were 
originally encumbered. As you well know, it was Heaviside who 
most strongly advocated the banishing of these superfluous factors and 
it will be well, I think, to follow his advice. Our unit of electricity 
will therefore be Y^x times smaller than the usual electrostatic unit. 



1) The units and the notation of these lectures (with the exception of the 
letters serving to indicate vectors) have also been used in my articles on 
Maxwell's Theory and the Theory of Electrons, in the „Enoyklopadie der 
naathematischen Wissenschaften", Vol. V, 13 and 14. 



MATHEMATICAL NOTATION. 3 

This choice having been made, we have at the same time fixed for 
every case the number by which the electric force is to be represented. 
As to the magnetic force, we continue to understand by it the force 
acting on a north pole of unit strength; the latter however is like- 
wise y^Tt times smaller than the unit commonly used. 

2. Before passing on to the electromagnetic equations, it will be 
necessary to say a few words about the choice of the axes of coor- 
dinates and about our mathematical notation. In the first place, we 
shall always represent a line by s, a surface by (3 and a space by S, 
and we shall write ds, ds, dS respectively for an element of a line, 
a surface, or a space. In the case of a surface, we shall often have 
to consider the normal to it; this will be denoted by n. It is always 
to be drawn towards a definite side and we shall agree to draw it 
towards the outside, if we have to do with a closed surface. 

The normal may be used for indicating the direction of a 
rotation in the surface. We shall say that the direction of a rotation 
in a plane and that of a normal to the plane correspond to each 
other, if an ordinary or right-handed screw turned in the direction 
of the rotation advances in that of the normal. This being agreed 
upon, we may add that the axes of coordinates wiU be chosen in 
such a manner that OZ corresponds to a rotation of 90" from OX 
towards OY. 

We shall further find it convenient to use a simple kind of 
vector analysis and to distinguish vectors and scalar quantities by 
different sorts of letters. Conforming to general usage, I shall denote 
scalars by ordinary Latin or Greek letters. As to the vectors, I have, 
in some former publications, represented them by German letters. 
On the present occasion however, it seems to me that Latin letters, 
either capital or small ones, of the so called Clarendon type, e. g. 
A, P, C etc. are to be preferred. I shall denote by A^ the component 
of a vector A in the direction /*, by A^, A^^, A. its components parallel 
to the axes of coordinates, by A^ the component in the direction of 
a line s and finally by A„ that along the normal to a surface. 

The magnitude of a vector A will be represented by [ A | . For 
its square however we shall simply write A^. 

Of the notions that have been introduced into vector analysis, 
I must recall to your minds those of the sum and of the difference 
of vectors, and those of the scalar product and the vector product of 
two vectors A and B. The first of these „products", for which we 

shall use the symbol 

(A • B), 

is the scalar quantity defined by the formula 

(A . B) = 1 A I I B I cos (A, B) = A^B, + A^B^ + A,B,. 

1* 



4 I. GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 

The vector product, for which we shall write 

[A-B], 

is a vector perpendicular to the plane through A and B, whose 
direction corresponds to a rotation by less than 180" from the direc- 
tion of A towards that of B, and whose magnitude is given by the 
area of the parallelogram described with A and B as sides. Its 
components are 

[A B], = A^B, - A^B^, [A B]^ = A,B, - A,B,, 

[A.B1 = A,B^-A^B,. 

In many cases we have to consider a scalar quantity qo or a 
vector A which is given at every point of a certain space. If g? is a 
continuous function of the coordinates, we can introduce the vector 
having for its components 

dq) d<f df 

dx' dy' dz 

This can easily be shown to be perpendicular to the surface 

(p = const. 

and we may call it the gradient of tp, which, in our formulae, we 
shall shorten to „grad qp". 

A space at every point of which a vector A has a definite 
direction and a definite magnitude may be called a vector field, and 
the lines which at every point indicate the direction of A may be 
spoken of as vector- or direction-lines. In such a vector field, if 
A^, Ay, A^ are continuous functions of the coordinates, we can intro- 
duce for every point a certain scalar quantity and a certain new 
vector, both depending on the way in which A changes from point 
to point, and both having the property of being independent of the 
choice of the axes of coordinates. The scalar quantity is called the 
(Uver(jence of A and defined by the formula 

div A = -„-- + -^- -f -^^ • 
ox dy OS 

The vector is called the rotation or the curl of A; its com- 
ponents are 

aA, _d_ky 2A^ _ 3A^ ?Ay _ ^ 

dy dz' dz dx > dx dy ' 

and it will be represented by the symbol „rot A". 

If the divergence of a vector is at all points, its distribution 
over space is said to be solenoidal. On the other hand, we shall 
speak of an irrotational distribution, if at all points we have 
rot A = 0. 



FUNDAMENTAL EQUATIONS FOR THE ETHER. 5 

In order to complete our list of notations, I have only to add 
that the symbol A is an abbreviation for 

and that not only scalars but also vectors may be differentiated with 

dx 



3A 
respect to the coordinates or the time. For example, ^ means a 



vector whose components are 

a_A^_ 3 Ay aA^ 

dx ' Bx ' dx ' 

SA 
and -^ has a similar meaning. A differentiation with respect to the 

time t will be often represented by a dot, a repeated differentiation 
of the same kind by two dots, etc. 

3. We are now prepared to vrrite down the fundamental equa- 
tions for the electromagnetic field in the form which they take for 
the ether. We shall denote by d the electric force, the same symbol 
serving for the dielectric displacement, because in the ether this has 
the same direction and, on account of the choice of our units, the 
same numerical magnitude as the electric force. We shall further 
represent by h the magnetic force and by c a constant depending on 
the properties of the ether. A third vector is the current C, which 
now consists only of the displacement current of Maxwell. It exists 
wherever the dielectric displacement d is a function of the time, and 
is given by the formula 

C = d. (1) 

In the form of differential equations, the formulae of the electro- 
magnetic field may now be written as follows: 

div d = 0, (2) 

div h = 0, (3) 

roth = -^c = yd, (4) 

rot d = - -^ h. (5) 

The third equation, conjointly with the second, determines the 
magnetic field that is produced by a given distribution of the 
current C. As to the last equation, it expresses the law according 
to which electric forces are called into play in a system with a 
variable magnetic field, i. e. the law of what is ordinarily called 
electromagnetic induction. The formulae (1), (4) and (5) are vector 
equations and may each be replaced by three scalar equations relating 
to the separate axes of coordinates. 



6 I. GENERAL PBUSrCIPLES. THEORY OF FREE ELECTRONS. 

Thus (1) is equivalent to 



and (4) to 



•^^"""27' '^ dt> ^^~ dt ' 



dy dz c dt ' 
The state of things that is represented by our fundamental 
equations consists, generally speaking, in a propagation with a velo- 
city c. Indeed, of the six quantities d^, d^, d,, h^, h^, h,, five may 
be eliminated 1), and we then find for the remaining one ^ an equation 
of the form 

A.-i.|? = 0. (6) 

This is the typical differential equation for a disturbance of the 
state of equilibrium, travelling onvrards with the speed c. 

Though all the solutions of our equations have this general 
character, yet there are a very large variety of them. The simplest 
corresponds to a system of polarized plane waves. For waves of this 
kind, we may have for example 

i^^ = a cos n(t--~y h^ = a cos n(t-^j, (7) 

all other components of d and h being 0. 

I need not point out to you that really, in the state represented 
by these formulae, the values of d^, and h., which for a certain value 
of t exist at a point with the coordinate x, will after a lapse of 
time dt be found in a point whose coordinate is a; + cSt. The 
constant a is the amplitude and n is the frequency, i. e. the number 
of vibrations in a time 2?!;. If n is high enough, we have to do 
with a beam of plane polarized light, in which, as you know already, 
the electric and the magnetic vibrations are perpendicular to the ray 
as well as to each other. 

Similar, though perhaps much more complicated formulae may 
serve to represent the propagation of Hertzian waves or the radiation 
which, as a rule, goes forth from any electromagnetic system that is 
not in a steady state. If we add the proper boundary conditions, 
such phenomena as the diffraction of light by narrow openings or 
its scattering by small obstacles may likewise be made to fall under 
our system of equations. 

The formulae for the ether constitute the part of electromagnetic 
theory that is most firmly established. Though perhaps the way in 
which they are deduced will be changed in future years, it is 



1) See Note 1 (Appendix). 



GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD. 7 

hardly conceivable tliat the equations themselves virill have to he 
altered. It is only vrhen we come to consider the phenomena in 
ponderable bodies, that we are led into uncertainties and doubts. 

4. There is one way of treating these phenomena that is compa- 
ratively safe and, for many purposes, very satisfactory. In following 
it, we simply start from certain relations that may be considered as 
expressing, in a condensed form, the more important results of electro- 
magnetic experiments. We have now to fix our attention on foiir 
vectors, the electric force E, the magnetic force H, the current of 
electricity C and the magnetic induction B. These are connected by 
the following fundamental equations: 

div C = 0, (8) 

div B = 0, (9) 

rotH=4c, (10) 

rotE = -iB, (11) 

presenting the same form as the formulae we have used for the ether. 
In the present case however, we have to add the relation between 
E and C on the one hand, and that between H and B on the other. 
Confining ourselves to isotropic bodies, we can often describe the 
phenomena with sufficient accuracy by writing for the dielectric dis- 
placement 

D = aE, (12) 

a vector equation which expresses that the displacement has the same 
direction as the electric force and is proportional to it. The current 
in this case is again Maxwell's displacement current 

C = D. : f (13) 

In conducting bodies on the other hand, we have to do with a 
current of conduction, given by 

J = (jE, (14) 

where ^ is a new constant. This vector is the only current and 
therefore identical to what we have called C, if the body has only 
the properties of a conductor. In some cases however, one has been 
led to consider bodies endowed with the properties of both conductors 
and dielectrics. If, in a substance of this kind, an electric force is 
supposed to produce a dielectric displacement as well as a current 
of conduction, we may apply at the same time (12) and (14), writing 
for the total current 

C = D + J = £E + 0E. (15) 



8 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

Finally, the simplest assumption we can make as to the relation 
between the magnetic force and the magnetic induction is expressed 
by the formula 

B = fxH, (16) 

in which fi. is a new constant. 

5. Though the equations (12), (14) and (16) are useful for the 
treatment of many problems, they cannot be said to be applicable to 
all cases. Moreover, even if they were so, this general theory, in 
which we express the peculiar properties of different ponderable 
bodies by simply ascribing to each of them particular values of the 
dielectric constant b, the conductivity <S and the magnetic permeabi- 
lity ft, can no longer be considered as satisfactory, when we wish to 
obtain a deeper insight into the nature of the phenomena. If we 
want to understand the way in which electric and magnetic properties 
depend on the temperature, the density, the chemical constitution or 
the crystalline state of substances, we cannot be satisfied with simply 
introducing for each substance these coefficients, whose values are 
to be determined by experiment; we shall be obliged to have recourse 
to some hypothesis about the mechanism that is at the bottom of 
the phenomena. 

It is by this necessity, that one has been led to the conception 
of electrons, i. e. of extremely small particles, charged with electricity, 
which are present in immense numbers in all ponderable bodies, and 
by whose distribution and motions we endeavor to explain all electric 
and optical phenomena that are not confined to the free ether. My 
task will be to treat some of these phenomena in detail, but I may 
at once say that, according to our modern views, the electrons in 
a conducting body, or at least a certain part of them, are supposed 
to be in a free state, so that they can obey an electric force by 
which the positive particles are driven in one, and the negative 
electrons in the opposite direction. In the case of a non-conducting 
substance, on the contrary, we shall assume that the electrons are 
bound to certain positions of equilibrium. If, in a metallic wire, the 
electrons of one kind, say the negative ones, are travelling in one 
direction, and perhaps those of the opposite kind in the opposite 
direction, we have to do with a current of conduction, such as may 
lead to a state in which a body connected to one end of the wire 
has an excess of either positive or negative electrons. This excess, 
the charge of the body as a whole, will, in the state of equilibrium 
and if the body consists of a conducting substance, be found in a 
very thin layer at its surface. 

In a ponderable dielectric there can likewise be a motion of the 



ELECTRONS. 9 

electrons. Indeed, though we shall think of each of them as having 
a definite position of equilibrium, we shall not suppose them to be 
wholly immoYable. They can be displaced by an electric force exerted 
by the ether, which we conceive to penetrate all ponderable matter, 
a point to which we shall soon have to revert. Now, however, the 
displacement will immediately give rise to a new force by which the 
particle is pulled back towards its original position, and which we may 
therefore appropriately distinguish by the name of elastic force. The 
motion of the electrons in non-conducting bodies, such as glass and 
sulphur, kept by the elastic force within certain bounds, together 
with the change of the dielectric displacement in the ether itself, 
now constitutes what Maxwell called the displacement current. 
A substance in which the electrons are shifted to new positions is 
said to be electrically polarized. 

Again, under the influence of the elastic forces, the electrons can 
vibrate about their positions of equilibrium. In doing so, and perhaps 
also on account of other more irregular motions, they become the 
centres of waves that travel outwards in the surrounding ether and 
can be observed as light if the frequency is high enough. In this 
manner we can account for the emission of light and heat. As to 
the opposite phenomenon, that of absorption, this is explained by 
considering the vibrations that are communicated to the electrons 
by the periodic forces existing in an incident beam of light. If the 
motion of the electrons thus set vibrating does not go on undisturbed, 
but is converted in one way or another into the irregular agitation 
which we call heat, it is clear that part of the incident energy will 
be stored up in the body, in other terms that there is a certain ab- 
sorption. Nor is it the absorption alone that can be accounted for 
by a communication of motion to the electrons. This optical resonance, 
as it may in many cases be termed, can likewise make itself felt 
even if there is no resistance at all, so that the body is perfectly 
transparent. In this case also, the electrons contained within the 
molecules will be set in motion, and though no vibratory energy is 
lost, the oscillating particles will exert an influence on the velocity 
with which the vibrations are propagated through the body. By 
taking account of this reaction of the electrons we are enabled to 
establish an electromagnetic theory of the refrangibility of light, in 
its relation to the wave-length and the state of the matter, and to 
form a mental picture of the beautiful and varied phenomena of 
double refraction and circular polarization. 

On the other hand, the theory of the motion of electrons in 
metallic bodies has been developed to a considerable extent. Though 
here also much remains to be done, new questions arising as we 
proceed, we can already mention the important results that have 



10 I. GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 

been reached by Riecke, Drude and J. J. Thomson.^) The funda- 
mental idea of the modern theory of the thermic and electric pro- 
perties of metals is, that the free electrons in these bodies partake 
of the heat-motion of the molecules of ordinary matter, travelling in 
all directions with such velocities that the mean kinetic energy of 
each of them is equal to that of a gaseous molecule at the same 
temperature. If we further suppose the electrons to strike over and 
over again against metallic atoms, so that they describe irregular 
zigzag-lines, we can make clear to ourselves the reason that 
metals are at the same time good conductors of heat and of electri- 
city, and that, as a general rule, in the series of the metals, the two 
conductivities change in nearly the same ratio. The larger the 
number of free electrons, and the longer the time that elapses between 
two successive encounters, the greater wiU be the conductivity for 
heat as well as that for electricity. 

6. This rapid review will suffice to show you that the theory 
of electrons is to be regarded as an extension to the domain of 
electricity of the molecular and atomistic theories that have proved 
of so much use in many branches of physics and chemistry. Like 
these, it is apt to be viewed unfavourably by some physicists, who 
prefer to push their way into new and unexplored regions by follow- 
ing those great highways of science which we possess in the laws 
of thermodynamics, or who arrive at important and beautiful results, 
simply by describing the phenomena and their mutual relations by 
means of a system of suitable equations. No one can deny that 
these methods have a charm of their own, and that, in following 
them, we have the feeling of treading on firm ground, whereas in 
the molecular theories the too adventurous physicist often runs the 
risk of losing his way and of being deluded by some false prospect 
of success. We must not forget, however, that these molecular hypo- 
theses can boast of some results that could never have been attained 
by pure thermodynamics, or by means of the equations of the electro- 
magnetic field in their most general form, results that are well known 
to all who have studied the kinetic theory of gases, the theories of 



1) E. Rieoke, Zur Theorie des Galvanismus und der Warme, Ann. Ptys. 
Chem. 66 (1898), p. 353, 545, 1199; Uber das Verhaltnis der Leitfiihigkeiten 
der Metalle fiir Waxme und fiir Elektrizitat, Ann. Phys. 2 (1900), p. 835. 
P. Drude, Zur Elektronentlieorie der Metalle, Ann. Phys. 1 (1900), p. 566; 
3 (1900), p. 369. J. J. Thomson, Indications relatives a la constitution de la 
matifere fournies par les reoherches recentes sur le passage de IMlectricit^ a 
travers les gaz, Rapports du Congres de physique de 1900, Paris, 3, p. 138. 
See also H. A. Lorentz, The motion of electrons in metallic bodies, Amsterdam 
Proc. 1904—1905, p. 438, 588, 684. 



ELECTRONS AND ETHER. 11 

dilute solutions, of electrolysis and of the genesis of electric currents 
by the motion of ions. Nor can the fruitfulness of these hypotheses 
be denied by those who have followed the splendid researches on the 
conduction of electricity through gases of J. J. Thomson^) and his 
fellow workers. 

7. I have now to make you acquainted with the equations 
forming the foundation of the mathematical theory of electrons. 
Permit me to introduce them by some preliminary remarks. 

In the first place, we shall ascribe to each electron certain finite 
dimensions, however small they may be, and we shall fix our attention 
not only on the exterior field, but also on the interior space, in 
which there is room for many elements of volume and in which the 
state of things may vary from one point to another. As to this 
state, we shall suppose it to be of the same kind as at outside points. 
Indeed, one of the most important of our fundamental assumptions 
must be that the ether not only occupies all space between molecules, 
atoms or electrons, but that it pervades all these particles. We shall 
add the hypothesis that, though the particles may move, the ether 
alivays remains at rest. We can reconcile ourselves with this, at 
first sight, somewhat startling idea, by thinking of the particles of 
matter as of some local modifications in the state of the ether. These 
modifications may of course very well travel onward while the volume- 
elements of the medium in which they exist remain at rest. 

Now, if within an electron there is ether, there can also be an 
electromagnetic field, and all we have got to do is to establish a 
system of equations that may be applied as well to the parts of the 
ether where there is an electric charge, i. e. to the electrons, as to 
those where there is none. As to the distribution of the charge, we 
are free to make any assumption we like. For the sake of convenience 
we shall suppose it to be distributed over a certain space, say over 
the whole volume occupied by the electron, and we shall consider 
the volume-density p as a continuous function of the coordinates, so 
that the charged particle has no sharp boundary, but is surrounded 
by a thin layer in which the density gradually sinks from the value 
it has within the electron to 0. Thanks to this hypothesis of the 
continuity of q, which we shall extend to all other quantities occurring 
in our equations, we have never to trouble ourselves about surfaces 
of discontinuity, nor to encumber the theory by separate equations 
relating to these. Moreover, if we suppose the difference between 
the ether within and without the electrons to be caused, at least so 



1) J. J. Thomson, Conduction of electricity througli gases, Cambridge, 
1903. 



12 I. GENERAL PRINCIPLBS. THEORY OF FREE ELECTRONS. 

far as we are concerned with it, only by the existence of the volume- 
density in the interior, the equations for the external held must be 
got from those for the internal one by simply putting p = 0, so that 
we have only to write down one system of differential equations. 

Of course, these must be obtained by a suitable modification, in 
which the intluence of the charge is expressed, of the equations 
(2) — (5) which we have established for the free, i. e. for the uncharged 
ether. It has been found that we can attain our object by the 
slightest modification imaginable, and that we can assume the following 
system 

div d = 9, (17) 

div h = 0, (18) 

roth = ic = -(ti + pv), (19) 



rotd=-4h, (20) 

in which the first and the third formula are the only ones that have 
been altered. 

In order to justify these modifications, I must in the first place 
recall to your minds the general relation existing in Maxwell's 
theory between the dielectric displacement across a closed surface 
and the amount of charge e contained within it. It is expressed by 
the equation 

fd^d6 = e, (21) 

in which the integral relates to the closed surface, each element d0 
of it being multiplied by the component of d along the normal n, 
which, as we have already said, is drawn towards the outside. Using 
a well known form of speech and comparing the state of things with 
one in which there would be no dielectric displacement at all, we 
may say that the total quantity of electricity that has been displaced 
across the surface (a quantity that has been shifted in an outward 
direction being reckoned as positive), is equal to the charge e. Now, 
if we apply this to an element of space dxdydz, taken at a point 
where there is a volume-density p, we have 

e = Qdxdydz 

and, since the integral in (21) reduces to 

div A • dxdy ds, 

we are at once led to the formula (17). 

In the second place, we must observe that a moving charge 
constitutes what is called a convection current and produces the 
same magnetic effects as a common current of conduction; this was 



FUNDAMENTAL EQUATIONS. 13 

first shown by Rowland's celebrated and well known experiment. 
Now, if V is the Telocity of the charge, it is natural to write pv for 
the convection current; indeed, the three components q\^, qV , q\^ 
represent the amounts of charge, reckoned per unit of area and unit 
of time, which are carried across elements of surface perpendicular to 
the axes of coordinates. On the other hand, if in the interior of an 
electron there is an electromagnetic field, there will also be a 
displacement current d. We are therefore led to assume as the 
expression for the total current 

c = d + 9V, (22) 

and to use the equation (19) in order to determine the magnetic 
field. Of course, this is again a vector equation. In applying it to 
special problems, it is often found convenient to replace it by the 
three scalar differential equations 

dh, chy 1 /ad^ , ... \ dK 2h, 1 /ad^ 



dy 8z 



e \dt T^^W' Be dx c\dt +?%j' 



c\dt '^ PW' 



dx dy 

You see that by putting 9 = 0, in the formulae (17) and (19), 
we are led back to our former equations (2) and (4). 

8. There is one more equation to be added, in fact one that 
is of equal importance with (17) — (20). It will have been noticed 
that I have carefully abstained from saying anything about the 
nature of the electric charge represented by q. Speculations on this 
point, or attempts to reduce the idea of a charge to others of a 
different kind, are entirely without the scope of the present theory; 
we do not pretend to say more than this, that p is a quantity, 
belonging to a certain point in the ether and connected with the 
distribution of the dielectric displacement in the neighbourhood of 
that point by the equation (17). We may say that the ether can 
be the seat of a certain state, determined by the vector d which we 
call the dielectric displacement, that in general this vector is solenoidally 
distributed, but that there are some places which form an exception 
to this rule, the divergence of d having a certain value q, different 
from 0. In such a case, we speak of an electric charge and under- 
stand by its density the value of div d. 

As to 'the statement that the charges can move through the 
ether, the medium itself remaining at rest, if reduced to its utmost 
simplicity, it only means that the value of div d which at one moment 
exists at a point P, will the next moment be found at another 
place P'. 



14 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

Yet, in order to explain electromagnetic phenomena, we are 
obliged to go somewhat further. It is not quite sufficient to con- 
sider Q as merely the symbol for a certain state of the ether. On 
the contrary, we must invest the charges with a certain degree ot 
substantiality, so far at least that we recognize the possibility of 
forces acting on them and producing or modifying their motion, ihe 
word „force" is here taken in the ordinary sense it has in dynamics, 
and we should easily become accustomed to the idea of forces acting 
on the charges, if we conceived these latter as fixed to what we are 
accustomed to call matter, or as being a property of this matter. 
This is the idea underlying the name of „charged particle" which 
we have already used and shall occasionally use again for an electron. 
We shall see later on that, in some cases at least, the fitness of the 
name is somewhat questionable. 

However this may be, we must certainly speak of such a thing 
as the force acting on a charge, or on an electron, on charged 
matter, whichever appellation you prefer. Now, in accordance with 
the general principles of Maxwell's theory, we shall consider 
this force as caused by the state of the ether, and even, since 
this medium pervades the electrons, as exerted by the ether on all 
internal points of these particles where there is a charge. If we 
divide the whole electron into elements of volume, there will be a 
force acting on each element and determined by the state of the 
ether existing within it. We shall suppose that this force is pro- 
portional to the charge of the element, so that we only want to 
know the force acting per unit charge. This is what we can now 
properly call the electric force. We shall represent it by f. The 
formula by which it is determined, and which is the one we still 
have to add to (17) — (20), is as follows: 

f = cl-fl[v.h]. (23) 

Like our former equations, it is got by generalizing the results of 
electromagnetic experiments. The first term represents the force 
acting on an electron in an electrostatic field; indeed, in this case, 
the force per unit of charge must be wholly determined by the 
dielectric displacement. On the other hand, the part of the force 
expressed by the second term may be derived from the law according 
to which an element of a wire carrying a current is acted on by a 
magnetic field with a force perpendicular to itself and the lines of 
force, an action, which in our units may be represented in vector 
notation by 

F = |[i-h], 



FORCE ACTING ON UNIT CHARGE. 15 

where i is tlie intensity o£ the current considered as a vector, and s 
the length of the element. According to the theory of electrons, 
F is made up of all the forces with which the field h acts on the 
separate electrons moving in the wire. Now, simplifying the question 
by the assumption of only one kind of moving electrons with equal 
charges e and a common velocity V, we may write 

si = iVev, 

if N is the whole number of these particles in the element s. Hence 

F = ^[v.h], 

so that, dividing by Ne, we find for the force per unit charge 

1 



c 



[V . h]. 



As an interesting and simple application of this result, I may mention 
the explanation it affords of the induction current that is produced 
in a wire moving across the magnetic lines of force. The two kinds 
of electrons having the velocity V of the wire, are in this case driven 
in opposite directions by forces which are determined by our formula. 

9. After having been led in one particular case to the existence 

of the force d, and in another to that of the force — [v ■ h], we now 

combine the two in the way shown in the equation (23), going 
beyond the direct result of experiments by the assumption that in 
general the two forces exist at the same time. If, for example, an 
electron were moving in a space traversed by Hertzian waves, we 
could calculate the action of the field on it by means of the values 
of d and h, such as they are at the point of the field occupied by 
the particle. 

Of course, in cases like this, in which we want to know the 
force exerted by an external field, we need not distinguish the 
directions and magnitudes of f at different points of the electron, at 
least if there is no rotation of the particle; the velocity V will be 
the same for all its points and the external field may be taken as 
homogeneous on account of the smallness of the electron. If however, 
for an electron having some variable motion, we are required to 
calculate the force that is due to its own field, our analysis must be 
pushed further. The field is now far from homogeneous, and after 
having divided the particle into elements of volume, we must 
determine the action of the field on each of them. Finally, if the 
electron is treated as a rigid body, we shall have to calculate in the 
ordinary way the resultant force and the resultant couple. 



16 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

10. While I am speaking so boldly of what goes on in the 
interior of an electron, as if I had been able to look into these small 
particles, I fear one will feel inclined to think I had better not try- 
to enter into all these details. My excuse must be that one can 
scarcely refrain from doing so, if one wishes to have a perfectly 
definite system of equations; moreoyer, as we shall see later on, our 
experiments can really teach us something about the dimensions oi 
the electrons. In the second place, it may be observed that in those 
cases in which the internal state of the electrons can make itself 
felt, speculations like those we have now entered upon, are at all 
events interesting, be they right or wrong, whereas they are harm- 
less as soon as we may consider the internal state as a matter of 
little importance. 

It must also be noticed that our assumptions by no means 
exclude the possibility of certain distributions of charge which we 
have not at first mentioned. By indefinitely diminishing' the thickness 
of the transition layer in which q passes from a finite value to 0, 
we can get as a limiting case that of an electron with a sharp 
boundary. We can also conceive the charge to be present, not 
throughout the whole extent of the particle, but only in a certain 
layer at its surface, whose thickness may be made as small as we 
like, so that after aU we can speak of a surface -charge. Indeed, in 
some of our formulae we shall have in view this special case. 

11. Since our equations form the real foundation-stones of the 
structure we are going to build, it will be weU to examine them 
somewhat more closely, so that we may be sure that they are con- 
sistent with each other. They are easily shown to be so, provided 
only the charge of an element of volume remain constant during its 
motion.^) If we regard the electrons as rigid bodies, as we shall 
almost always do, this of course means that q is constant at every 
point of a particle. However, we might also suppose the electrons to 
change their shape and volume; only, in this case, the value of q 
for an element of volume ought to be considered as varying in the 
inverse ratio as the magnitude of the element. 

It is also important to remark that our formulae are applicable 
to a system in which the charges, instead of being concentrated in 
certain small particles, are spread over larger spaces in any way you 
like. We may even go a step further and imagine any number of 
charges with the densities q^ , q^ etc., which are capable of penetrating 
each other and therefore of occupying the same part of space, and 
which move, each with its own velocity. This would require us to 

1) Note 2. 



DETERMINATION OF THE FIELD. 17 

replace the terms q and ^v in (17) and (19) by Pi + Ps H and 

Qi^i + Q-i^^-i , the vectors Vj, v^, ... being the velocities of the 

separate charges. An assumption of this kind, artificial though it 
may seem, will be found of use in one of the problems we shall 
have to examine. 

12. I have now to call your attention to some of the many 
beautiful results that may be derived from our fundamental equations, 
in the first place to the way in which the electromagnetic field is 
determined by the formulae (17)— (20), if the distribution and the 
motion of the charges are supposed to be given. The possibility of 
this determination is due to the fact that we can eliminate five of 
the six quantities d^, d^,, d^, h^, h^^, h^, exactly as we could do, when 
we treated the equations for the free ether, and to the remarkable 
form in which the final equation presents itself^) We have, for 
example, three equations for the components of d, which we may 
combine into the vector formula 

A d - i, d = grad Q + ^^Yt (P^)' (24) 

and the similar condition for the magnetic force 

Ah-i,h= -irot(pv). (25) 

It will not be necessary to write down the six scalar equations for 
the separate components; we can confine ourselves to the formulae 
for d^ and h^, viz. 

Ad.-if,-|| + i?^^, (26) 

Ah.-ih; = -i(?te>-^). (27) 

In order to express myself more clearly, it will be proper to 

introduce a name for the left-hand sides of these equations. The 

result of the operation A, applied to a quantity if that is a function 

of the coordinates x, y, z, has been called the Laplacian of i^. 

1 8^ 
Similarly, the result of the operation A ^ -i^ may be given the name 

of the Dalembertian of the original quantity, in commemoration 
of the fact that the mathematician d'Alembert was the first to 
solve a partial differential equation, occurring in the theory of a 
vibrating string, which contains this operation, or rather the operation 

^— I 2 -„- 5, which is a special case of it. Of course, since vectors 

can be differentiated with respect to time and place, we may as well 



1) Note 3. 

Lorentz, Theory of electrons. 



18 I. GENERAL PRmCIPLES. THEORY OF FREE ELECTRONS. 

speak of tlie Dalerabertian of a vector as of that of a scalar quan- 
tity. Accordingly, since, for a given distribution and motion of the 
charges, the right-hand members of our last equations are known 
functions of x, y, s, t, we see that the vectors d and h, as well as 
each of their components, are determined by the values of their 
Dalembertians. We have therefore to look into the question, what 
will be the value of a quantity il) whose Dalembertian has a 
given value co. This is a problem which admits a simple solution. 
In the ordinary theory of the potential it is proved that a function t^ 
whose Laplacian has a given value a, may be found by the 
formula 

^ iTtJ r 

where r is the distance from an element of volume dS to the point P 
for which we want to calculate ip, a the value of the Laplacian 
in this element, and where we have to integrate over all parts of 
space in which o is different from 0. 

Now, it is very remarkable that a function i^ satisfying the 
equation 

may be found by a calculation very like that indicated in (28)^). The 
only difference is that, if we are asked to determine the value of ^ 
at the point P for the instant t, we must take for a the value of 

this function existing in the element dS at the time t • We 

shall henceforth include in square brackets quantities whose values 

must be taken, not for the time t, but for the previous time t 

Using this notation, we may say that the function 

is a solution of the differential equation (29). It should be observed 

that this also holds when cs is a vector quantity; [to] and ~dS 

will then be so likewise, and the integration in (30) is to be under- 
stood as the addition of an infinite number of infinitely small vectors. 
For purposes of actual computation, the vector equation may again 
be split up into three scalar ones, containing the components of m, 
and giving us those of ^t. 

1) Note 4. 



POTENTIALS. 19 

13. The above method of calculation might be applied to the 
equations (24) and (25) or (26) and (27). Since, however, the second 
members of these formulae are somewhat complicated, we prefer not 
directly to determine d and h, but to calculate in the first place 
certain auxiliary functions on which the electric and magnetic forces 
may be made to depend, and which are called potentials. The first 
is a scalar quantity, which I shall denote by (p, the second a vector 
for which I shall write a. 

If the potentials are subjected to the relations 

and 

Aa-^a = -49V, (32) 

one can show^), by means of (17) — (20), that the dielectric displacement 
is given by 

d = a — grad (p (33) 

and the magnetic force by 

h = rot a. (34) 

You see that the equations (31) and (32) are again of the form (29), 
so that the two potentials are determined by the condition that their 

Dalembertians must have the simple values — q and q\. 

Therefore, on account of (30), we may write 

^ = ^J\mdS. (36) 

By these equations, combined with (33) and (34), our problem is 
solved. They show that, in order to calculate the field, we have to 
proceed as follows: Let P be the point for which we wish to 
determine the potentials at the time t. We must divide the whole 
surrounding space into elements of volume, any one of which is 
called dS. Let it be situated at the point Q and let the distance QP 
be denoted by r. In this element of space there may or may not 
be a part of an electron at a certain time. We are only concerned 

with the question whether it contains a charge at the time t -■ 

Indeed, the brackets serve to remind us that we are to understand 

1) Note 5. 

2* 



and 



20 I- GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 

r 

by p the density existing in dS at the particular instant t - 

and by p V the product of this density and the velocity of the charge 
within dS &i that same instant. These values [()] and [pv] must be 
multiplied \,^ d8 and divided by r. Finally, we have to do for all 
elements what we have done for the one d8 and to add all the 
results. Of course there may be many elements which do not con- 
tribute anything to the integrals, viz. all those which at the time 

t did not contain any charge. 

14. What has been said calls forth some further remarks. In 
the first place, you see that the factor j-, which we have been so 
anxious to get rid of, has again appeared. We cannot prevent it 
from doing so, but fortunately it is now confined to a few of our 
equations. In the second place, it is especially important to observe 
that the values of q and q\ existing in a certain point Q at the 

time t do not make themselves felt at the point P at the same 

moment t — ~, but at the later time t. We may therefore really 

speak of a propagation taking place with the velocity c. The parts 
of (f) and a which are due to the several elements dS correspond to 
states existing in these elements at times which are the more remote, 
the farther these elements are situated from the point P considered. 

On account of this special feature of our result, the potentials qo 
and a, given by (35) and (3(3), are often called retarded potentials. 

I must add that the function (30) is not the most general 
solution of (29), and that for this reason the values of (33) and (34) 
derived from (35) and (36) are not the only ones satisfying the 
fundamental equations. We need not however speak of other solutions, 
if we assume that an electromagnetic field in the ether is never pro- 
duced by any other causes than the presence and motion of electrons.') 

15. The case of a single electron furnishes a good example for 
the application of our general formulae. Let us suppose in the first 
place that the particle never has had nor will have any motion. 
Then we have a = 0, and since q is the same at aU. instants, the 
scalar potential is given by 

cp = ^ f^dS. 

The equations (33) and (34) becoming 

d = — grad (p, h = 0, 
we fall back on the ordinary formulae of electrostatics. 
1) Note 6. 



FIELD OF A MOVING ELECTRON. 21 

We shall next consider an electron having (from if = — oo until 
t = -\- oo) a translation with constant velocity w along a straight 
line. Let P and P' he two points in such positions that the line 
PP' is in the direction of the motion of the particle. It is easily 
seen that, if we wish to calculate qo, a, d and h, first for the point P 

and the time t, and then for the point P' and the time t -] , 

we shall have to repeat exactly the same calculations. If, for example, 
dS is an element of space contributing a part to the integrals (35) 
and (36) in the first problem, the corresponding integrals in the 
second will contain equal parts due to an element dS' which may 
be got by shifting dS in the direction of translation over a distance 
equal to PP'. 

It appears from this that the electron is continually surrounded 
by the same field, which it may therefore be said to carry along 
with it. As to the nature of this field, one can easily deduce from 
(33) — (36) that, in the case of a spherical electron with a charge 
symmetrically distributed around the centre, if s is the path of the 
centre, the electric lines of force are curves situated in planes 
passing through s, and the magnetic lines circles having s as axis.') 
The field is distinguished from that of an electron without translation, 
not only by the presence of the magnetic force, but also by an 
alteration in the distribution of the dielectric displacement. 

We shall finally take a somewhat less simple case. Let us 
suppose that, from i = — oo until a certain instant t^, the electron 
is at rest in a position A, and that, in a short interval of time 
beginning at t^, it acquires a velocity w which remains constant in 
magnitude and direction until after some time, in a short interval 
ending at the instant ifg, the motion is stopped. Let B be the final 
position in which the electron remains for ever afterwards. 

If P is any point in the surrounding ether, we can consider two 
distances l^ and l^, the first of which is the shortest of aU the lines 
drawn from P to the points of the electron in the position A, and 
the second the longest of all the lines joining P to the electron in 
the position JB. We shall suppose the interval t^ — t^ to be so long 

that ^2 + 4 X, + -; • 

It will be clear that in performing the calculation of qp and a, 

for the point P and for an instant previous to ti + -^, we shall get 

a result wholly independent of the motion of the electron. This 
motion can by no means make itself felt in P during this first 
period, which will therefore be characterized by the field belonging 

1) Note 7. 



22 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

to the immovable electron. A similar field will exist in P after the 

time t^ + -^, every influence that has been emitted by the particle 

while moving, having already, in its outward progress, passed over 
the point considered. 

Between i^i + ^ and ^ + -^ the field at F will be due to the 

moving electron. If we suppose the dimensions of the particle to be 
very small in comparison with the distances \, l^, and the velocity w 
to be acquired and lost in intervals of time much shorter than t^ — tx, 
we may be sure that during the larger part of the interval between 

tx-\- ^ and t^ + — the field at P will be what it would have been, 

had a constant velocity iv existed for ever. Of course, immediately 

after ^i + — and shortly before t^ + -^ it will be otherwise; then, 

there will be a gradual transition from one state of things to the 
other. It is clear also that these periods of transition, taken for 
different points P, will not be found to coincide. If S^, S^, Sg are 
parts of space at different distances from the line AB, S-^ being the 
most remote and Sg the nearest, it may very well be that, at some 
particular instant, S, is occupied by the field belonging to the elec- 
tron while at rest in the position A, S^ by the field of the moving 
electron, and Sg by the final field. 

16. Thus far we have only used the equations (17) — (20). 
Adding to these the formula (23) for the electric force, and supposing 
the forces of any other nature which may act on the electrons to be 
given, we have the means of determining, not only the field, but 
also the motion of the charges. For our purpose however, it is not 
necessary to enter here into special problems of this kind. We shall 
concentrate our attention on one or two general theorems holding for 
any system of moving electrons. 

In the first place, suitable transformations of the fundamental 
formulae lead to an equation expressing the law of conservation of 
energy.^) If we confine ourselves to the part of the system lying 
within a certain closed surface e, this equation has the form 

jQ(f ^,)dS + ^J^lfiA'+hyis] + cJ[d.h]Ja = 0, (37) 

which we shall now try to interpret. Since f is the force with which 
the ether acts on unit charge, QfdS wiU be the force acting on the 
element dS of the charge, and 

(QdSf-\) = Q(f \)dS 

1) Note 8. 



ELECTRIC AND MAGNETIC ENERGY. FLOW OF ENERGY. 23 

its work per unit of time. The first integi-al in (37) is thus seen 
to represent the work done by the ether on the electrons per unit 
of time. Combined with the work of other forces to which the 
electrons may be subjected, this term will therefore enable us to 
calculate the change of the kinetic energy of the electrons. 

Of course, if the ether does work on the electrons, it must lose 
an equivalent amount of energy, a loss for which a supply of energy 
from the part of the system outside the surface 6 may make up, or 
which may be accompanied by a transfer of energy to that part. 
We must therefore consider 

-Ktl' + h-)rfS (38) 

as the expression for the energy contained within an element of 
volume of the ether, and 



c/[d hlda (39) 



as that for an amount of energy that is lost by the system within 
the surface and gained by the surrounding ether. 

The two parts into which (38) can be divided may properly be 
called the electric and the magnetic energy of the ether. Reckoned 
per unit of volume the former is seen to be 

w, = id^ (40) 

and the latter 

^m = W"- (41) 

These values are equivalent to those that were given long ago by 
Maxwell. That the coefficients are -^ and not 2x or something of 
the kind, is due to the choice of our new units and will certainly 
serve to recommend them. 

As to the transfer of energy represented by (39), it must 
necessarily take place at the points of the surface <? itself, because 
our theory leaves no room for any action at a distance. Further, 
we are naturally led to suppose that the actions by which it is 
brought about are such that, for each element dd, the quantity 
c[d ■ h]„(i(> may be said to represent the amount of energy that is 
transmitted across this particular element. In this way we come to 
the conception, first formulated by Poynting'), of a mrrent or flow 
of energy. It is determined by the vector product of d and h, mul- 
tiplied by the constant c, so that we can write for it 

s = c[d • h], (42) 



1) J. H. Poynting, On the transfer of energy in the electromagnetic field, 
London Trans. 176 (1884), p. 343. 



24 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

the meaning being that, for any element d6, the amount of energy 
by which it is traversed, is given for unit of time and unit of area 
by the component S„ of the vector S along the normal to the 
element. 

17. It is interesting to apply the above results to the beam of 
polarized light represented by our equations (7). We find for the 
energy which it contains per unit of space 

|(d^ + hO = a^cos^»^(i-|), 

and for the flow of energy across a plane perpendicular to the axis 

of X 

cd„h, = CO? cos^ nit )• 

y ' \ c / 

The mean values of these expressions for a full period are 
and 

Indeed, by a weU known theorem, the mean value of cos^ n\t j 

is i- 

It is easily seen that the expression Ica^ may also be used for 
calculating the flow of energy during any lapse of time that is very 
long compared with a period. 

If the beam of light is laterally limited by a cylindrical surface 
whose generating lines are parallel to OX, as it may be if we 
neglect diffraction phenomena, and if a normal section has the area 27, 
the flow of energy across a section is given by ^ca^U. It is equal 
for any two sections and must indeed be so, because the amount of 
energy in the part of the beam between them remains constant. 

The case of a single electron having a uniform translation like- 
wise affords a good illustration of what has been said about the 
flow of energy. After having determined the internal and the external 
field by means of the formulae (33) — (36), we can deduce the total 
electromagnetic energy from (40) and (41). I shall later on have 
occasion to mention the result. For the present I shaU. only say 
that, considering the course of the electric and the magnetic lines of 
force, which intersect each other at right angles, we must conclude 
that there is a current of energy, whose general direction is that of 
the translation of the electron. This should have been expected, 
since the moving electron is constantly surrounded by the same field. 
The energy of this field may be said to accompany the particle in 
its motion. 



FLOW OP ENERGY. 25 

Other examples migM likewise show us how Poynting's theorem 
throws a clear light on many questions. Indeed, its importance can 
hardly be overestimated, and it is now difficult to recall the state 
of electromagnetic theory of some thirty years ago, when we had to 
do without this beautiful theorem. 

18. Before leaving this subject I will, with your permission, 
call attention to the question, as to how far we can attach a definite 
meaning to a flow of energy. It must, I believe, be admitted that, 
as soon as we know the mutual action between two particles or ele- 
ments of volume, we shall be able to make a definite statement as 
to the energy given by one of them to the other. Hence, a theory 
which explains things by making definite assumptions as to the 
mutual action of the parts of a system, must at the same time admit 
a transfer of energy, concerning whose intensity there can be no doubt. 
Yet, even if this be granted, we can easily see that in general it 
will not be possible to trace the paths of parts or elements of energy 
in the same sense in which we can follow in their course the ultimate 
particles of which matter is made up. 

In order to show this, I shall understand by P a particle or an 
element of volume and hj A, B, C, . . . , A', B', (7', ... a certain number 
of other particles or elements, between which and P there is some 
action resulting in a transfer of energy and, in accordance with what 
has just been said, I shall suppose these actions to be so far known 
that we can distinctly state what amount of energy is interchanged 
between any two particles. Let, for example, P receive from A,B,C,... 
the quantities a, b, c, . . . oi energy, and let it give to A', B', C, . . . 
the quantities a, V, c, . . . , gaining for itself a certain amount p. 
Then we shall have the equation 

a + b + c-\----=p + a' + h' + c+---. 

Now, though in our imaginary case each term in this equation would 
be known, we should have no means for determining in what way 
the quantities of energy contained in a, b, c, . . , say the individual 
units of energy , are distributed among p, a, V, c, .... If, for 
example, there are only two terms on each side of the equation, 
all of the same value, so that it takes the form 

« + & = «' + &', 

we can neither conclude that a is the same energy as a and b' the 
same as b, nor that a is identical to b, and b' to a. There would 
be no means of deciding between these two views and others that 
likewise suggest themselves. 

For this reason, the flow of energy can, in my opinion, never 
have quite the same distinct meaning as a flow of material particles, 



26 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

which, by our imagination at least, we can distinguish from each 
other and follow in their motion. It might even be questioned 
whether, in electromagnetic phenomena, the transfer of energy really 
takes place in the way indicated by Poynting's law, whether, for 
example, the heat developed in the wire of an incandescent lamp is 
really due to energy which it receives from the surrounding medium, 
as the theorem teaches us, and not to a flow of energy along the 
wire itself In fact, all depends upon the hypotheses which we make 
concerning the internal forces in the system, and it may very well be, 
that a change in these hypotheses would materially alter our ideas 
about the path along which the energy is carried from one part of 
the system to another. It must be observed however that there is 
no longer room for any doubt, so soon as we admit that the pheno- 
mena going on in some part of the ether are entirely determined by 
the electric and magnetic force existing in that part. No one will 
deny that there is a flow of energy in a beam of light; therefore, 
if all depends on the electric and magnetic force, there must also be 
one near the surface of a wire carrying a current, because here, as 
well as in the beam of light, the two forces exist at the same time 
and are perpendicular to each other. 

19. Results hardly less important than the equation of energy, 
and of the same general character, are obtained when we consider 
the resultant of all the forces exerted by the ether on the electrons 
of a system. For this system we can take a ponderable body which 
is in a peculiar electromagnetic state or in which electromagnetic 
phenomena are going on. In our theory the ponderomotive force 
exerted on a charged conductor, a magnet or a wire carrying a 
current, is made up of all the forces with which the ether acts on 
the electrons of the body. 

Let 6 again be a closed surface, and F the resultant force on 
all the electrons contained within it. Then, on account of (23), we 
may write 

F=J(,{d+4[v.h]}rf,S', (43) 

extending the integral to all the electrons, or as we may do as well 
(p being in the space between the particles), to the whole space S. 
Now, by the aj)plication of the equations (17) — (20)^), this force F 
may be shown to be equal to the sum of two vectors 

F = F, + F„ (44) 

which are determined by the equations 

1) Note a. 



and 



STRESSES IN THE ETHER, 21 

^ix = \J[ 2d.d„ - d^ COS {n, x) } d0 
+ i / { 2h^h,, — h^ cos {n, x) ) d0 etc. 



(45) 



r, = -^J^dS. (46) 

The first part of the force is represented by an integral over 
the surface 6, its components, of which only one is given here, being 
determined by the values of d^, dj,, d^, h^, etc. at the surface. The 
second part of the force, on the contrary, presents itself as an integral 
over the space S, not only over those parts of it where there is an 
electric charge, but also over those where there is none. 

20. In discussing the above result we must distinguish several 
cases. 

a) In all phenomena in which the system is in a stationary 
state, the force F^, for which we may write 



F,= 



iS-J^dS, (47) 



disappears, and the whole force F is reduced to an integral over the 
surface 0. In other terms, the ponderomotive action can be regarded 
as the sum of certain infinitely small parts, each of which belongs 
to one of the surface-elements da and depends on the state existing 
at that element. A very natural way of interpreting this is to 
speak of each of these parts as of a stress in the ether, acting on 
the element considered. 

The stress depends on the orientation of the element. If this is 
determined by the normal n, and if, using a common notation, we 
write X^, Y^, Z^ for the components of the force per unit area, 
exerted by the part of the medium on the positive side of the sur- 
face on the part lying on the negative side, we shall have 

-^„ = i { 2 d,d„ - d^ cos in, a;) H- J. { 2 h,h„ - h^ cos {n, x) } etc. (48) 

From these formulae we can easily deduce the components X^, 
Y,, Z,, X„ etc. of the stresses acting on elements whose normal is 
parallel to one of tlie axes of coordinates. We find 

^. = i (d/ - d/ - tJ/) + ^- (h/ - K - ^') *^tc., (49) 

X^ = Z, = d,d^ + h,h^ etc., (50) 

precisely the values of the stresses by which Maxwell long ago 
accounted for the ponderomotive forces observed in electric and 
magnetic fields. 



28 



I. GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 



This method of calculating the resultant force is often very 
convenient, the more so because we can take for 6 any surface 
surrounding the body for which we have to solve the pi-oblem. 

b) We are led to similar conclusions if we consider a system 
that is the seat of periodical phenomena, confining ourselves to the 
mean value of the force taken for a complete period T. The mean 
value being given by 

T 



\J^dt, 



the last term in (44) disappears. Indeed, by (47) the time integral 
of Fj is equal to the diiference of the values of 



^J- 



sdS 



for i = and t = T, and these values are equal on account of the 
periodicity of the changes. 

Hence, in this case also, the resultant force is reduced to surface- 
integrals, or, as we may say, to stresses in the ether. 

It can easily be shown that the mean value of F (and of 
periodically changing quantities in general) during a lapse of time 
that is very much longer than a period T, is equal to the mean 
value during a period, even though the interval considered is not 
exactly a multiple of T. 



21. An interesting example is furnished by the pressure of 
radiation. Let (Fig. 1) AB he a. plane disk, receiving in a normal 

direction a beam of light L, which, taking 
OX in the direction shown in the diagram, 
we can represent by our formulae (7). Let 
us take for the surface of the flat cylin- 
drical box CDEF, whose plane sides lie 
before and behind the disk and are parallel 
to it. Then, if the plate is perfectly opaque, 
we have only to consider the stress on CD. 
Moreover, if the disk is supposed to be 
perfectly black, so that there is no reflected 
beam, there is only the electromagnetic field represented by the 
equations (7). Hence, since a normal to the plane CD, drawn 
towards the outside of the box CDEF, has a direction opposite 
to that of OX, the force acting on the absorbing body in the direc- 
tion of OX per unit area is given by 



^ 


' J 


iJC 


/- ^ 













X 



DBF 

Fig. 1. 



and its mean value by 



RADIATION PRESSURE. 29 



Comparing this with the value of the energy and attending to the 
direction of the force, we conclude that the beam of light produces 
a normal pressure on the absorbing body, the intensity of the pressure 
per unit of surface being numerically equal to the electromagnetic 
energy which ^the beam contains per unit of volume. 

The same method can be applied to a body which transmits 
and reflects a certain amount of light, and to a disk on which a 
beam of light falls in an oblique direction. In aU cases in which 
there is no light behind the disk, the force in the direction of the 
normal will be a pressure — X^ on the illuminated side, if the axis 
of X is directed as stated above. 

We shall apply this to a homogeneous and isotropic state of 
radiation, existing in a certain space that is enclosed by perfectly 
reflecting walls. By homogeneous and isotropic we mean that the 
space is traversed by rays of light or heat of various directions, in 
such a manner that the radiation is of equal intensity in diiferent 
parts of the space and in all directions, and that all directions of d 
and h are equally represented in it. It can easily be shown that in 
this case there is no tangential stress on an element ds oi the wall. 
As to the normal pressure, which is represented by — X^, if the 
axis of X is made to coincide with the normal, we may write for it 

P = 1(57+ d7- 6J) + UK + h? - K'), 

where the horizontal bars are intended to indicate the mean values, 
over the space considered, of the several terms. ^) But, on account 
of our assumptions regarding the state of radiation, 

d^ = d ^ = d^. 

Each of these quantities is therefore equal to one third of their sum, 
i. e. to 4-d^. Similarly 

Hence, if the formulae (40) and (41) are taken into account. 

In this case, the pressure on the walls per unit of surface is equal 
to one third of the electromagnetic energy per unit of volume. 

Later on, the problem of radiation pressure will be treated by 
a different method. 

1) Note 10. 



30 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

22. Thus far we have simplified the equation (44) by supposing 
the last term to vanish. In general, however, this term may not be 
omitted, and the force F cannot be accounted for by a system of 
stresses acting on the surface 6. 

This conclusion takes a remarkable form, if the surface 6 is 
supposed to enclose no electrons at all. Of course, the total force F 
must be in this case, as may be seen from the original expression (43). 
Nevertheless, the force due to the stresses is not generally 0, having 
the value 

Fi = -F, = Ajsc?S. (51) 

It is worthy of notice that this last equation is quite independent of 
the theory of electrons, being a consequence of the fundamental 
equations for the case Q = 0, i. e. of the equations for the free ether. 
It has indeed been known for a long time.^) 

In the mind of Maxwell and of many writers on the theory 
there seems to have been no doubt whatever as to the real existence 
of the ether stresses determined by the formulae (49) and (50). 
Considered from this point of view, the equation (51) tells us that 
in general the resultant force Fj of all the stresses acting on a part 
of the ether will not be 0. This was first pointed out by Helm- 
holtz.^) He inferred from it that the ether cannot remain at rest, 
and established a system of equations by which its motion can be 
determined. I shall not enter upon these, because no experiment has 
ever shown us any trace of a motion of the ether in an electro- 
magnetic field. 

We may sum up by saying that a theory which admits the 
existence of Maxwell's stresses leads to the following conclusions: 

1. A portion of the ether is not in equilibrium under the stresses 
acting on its surface. 

2. The stresses acting on the elements of a surface which 
surrounds a ponderable body will, in general, produce a resultant 
force different from the force acting on the electrons of the body 
according to our theory. 

23. Having got thus far, we may take two different courses. 
In the first place, bearing in mind that the ether is undoubtedly 
widely different from all ordinary matter, we may make the assump- 
tion that this medium, which is the receptacle of electromagnetic 
energy and the vehicle for many and perhaps for all the forces acting 
on ponderable matter, is, by its very nature, never put in motion, 

1) Note 11. 

2) Helmholtz, Polgerungen aus Maxwell's Theorie fiber die Bewegungen 
des reinen Athers, Ann. Phys. Chem. 63 (1894), p. 135. 



IMMOBILITY OF THE ETHER. 31 

that it lias neither velocity nor acceleration, so that we have no 
reason to speak of its mass or of forces that are applied to it. 
From this point of view, the action on an electron must be con- 
sidered as primarily determined by the state of the ether in the 
interior of each of its elements of volume, and the equation (43) as 
the direct and immediate expression for it. There is no reason at 
all why the force should be due to pressures or stresses in the uni- 
versal medium. If we exclude the idea of forces acting on the ether, 
we cannot even speak of these stresses, because they would be forces 
exerted by one part of the ether on the other. 

I should add that, while thus denying the real existence of 
ether stresses, we can still avail ourselves of all the mathematical 
transformations by which the application of the formula (43) may 
be made easier. We need not refrain from reducing the force to a 
surface-integral, and for convenience's sake we may continue to apply 
to the quantities occurring in this integral the name of stresses. 
Only, we must be aware that they are only imaginary ones, nothing 
else than auxiliary mathematical quantities. 

Perhaps all this that has now been said about the absolute 
immobility of the ether and the non-existence of the stresses, may 
seem somewhat startling. If it is thought too much so, one may 
have recourse to the other conception to which I have alluded. In 
choosing this, we recognize the real existence of Maxwell's internal 
forces, and we regard the ether as only approximately immovable. 

Let us admit that between adjacent parts of the ether there is 
an action determined by the equations (48), so that an element of 
volume of the free ether experiences a force 

—iidS, 

and let us suppose the medium to move in such a way that it has 
a momentum 

Is.?^, (52) 

or -^S per unit of volume. Let us further imagine that the density 

of the ether is so great that only a very small velocity, too small 
to be detected by any means at our disposal, is required for the 
momentum (52). Then, the formula (51) which, applied to an 
element of the ether, takes the form 

F, = —^sdS 

tells us that the assumed state of motion can really exist. This is 
clear because for very small velocities the resultant force acting on 



32 I. GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 

the ether contained in a fixed element of volume may be said to be 
equal to the rate of change of the momentum that is found wUhin 
that element.^) 

On the other hand, in the case of an element dS occupied by 
a charge, the formula 

? = ?,--,hd8 

may be interpreted as follows. The ether within the element is 
subject to a force F^, due to the stresses on the surface. Of this 
force, the part 

\kdS 

goes to produce the change of momentum of the ether, the remaining 
part F being transferred to the charge. 

You will readily perceive that, after all, the difference between 
the two modes of view consists mainly in the diiferent interpretations 
given to the same equations. 

24. Whatever may be our opinion about the questions we have 
now touched upon, our discussion shows the importance of the vector 

—iSdS, 

which has a definite direction and magnitude for every element of 
volume, and of the vector 



G 



^^fsdS (53) 



that maybe derived from it by integration. Abraham^) of Gottingen 
has applied to these quantities the name of electromagnetic momentum. 
We may term them so, even if we do not wish to convey the idea 
that they represent a real momentum, as they would according to 
the second of the two lines of thought we have just followed. 

The way in which the conception of electromagnetic momentum 
may be of use for the elucidation of electromagnetic phenomena 
comes out most clearly if, in dealing with a system of finite dimen- 
sions, as the systems in our experiments actually are, we make the 
enclosing surface recede on all sides to an infinite distance. It 
may be shown that the surface -integrals in (45) then become 0, so 
that, if the integration is extended to all space, we shall have 

f — % (54) 

1) Note 12. 

2) M. Abraham, Prinzipien der Dynamik des Elektrons, Ann. Phys. 10 
(1903), p. 105. 



RADIATION PRESSURE. 33 

or in words: the force exerted by the ether on a system of electrons, 
or, as we may say, on the ponderable matter containing these elec- 
trons, is equal and opposite to the change per unit of time of the 
electromagnetic momentum. Now, since the action tends to produce 
a change equal to the force itself in the momentum (in the ordinary 
sense of the word) of the ponderable matter, we see that the sum 
of this momentum and the electromagnetic one will not be altered 
by the actions exerted by the ether. 

Before passing on to one or two applications, I must call your 
attention to the intimate connexion between the momentum and the 
flow of energy S. The equation (53) at once shows us that every 
part of space in which there is a flow of energy contributes its part 
to the vector G; hence, in order to form an idea of this vector and 
of its changes, we have in the first place to fix our attention on the 
radiation existing in difi^erent parts of space. If, in course of time, 
the flow of energy reaches new parts of space or leaves parts in 
which it was at first found, this will cause the vector G to change 
from one moment to another. 

It must also be kept in mind that (53) is a vector equation and 
that (54) may be decomposed into three formulae giving us the 
components F^, F^,, F. of the resultant force. 

25. Very interesting illustrations of the preceding theory may 
be taken from the phenomena of radiation pressure, to which I shall 
therefore return for a moment. Let us consider, for example, a source 
of light sending out its rays in a single direction, which may be brought 
about by suitable arrangements, and let us suppose this radiation to 
have begun at a certain instant, so that we can speak of the first 
wave or of the front of the train of waves that have been emitted. 
This front is a plane at right angles to the beam and advancing 
with the velocity c. Hence, if U is the normal section of the beam, 
the volume occupied by the radiation increases hj cZ! per unit of 
time. As we have seen, the flow of energy has the direction of the 
beam. In making the following calculation, we shall reason as if, 
at every point, the flow were constantly equal to the mean flow S 
taken for a full period. If the magnitude of this mean flow, which 
relates to unit of area, is |8J, we shall flnd that of the electro- 
magnetic momentum, whose direction is likewise that of the beam, 

if we multiply ~2 S by the volume occupied by the light. It appears 

from this that the change of G per unit of time is 

I G I = - I S I £; 

II g I I 7 

consequently, since this vector has the direction of the rays, there 
will be a force on the source of light of the same intensity and in 

liorentz, Theory of electrons. 3 



34 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

a direction opposite to that in whicli the rays are emitted. This 
force of recoil, which, however, is extremely small, may be compared 
with the reaction that would exist if the rays of light consisted of 
a stream of material particles. By similar reasoning we can deter- 
mine the pressure on the black disk we have formerly considered. 
But, in this case, it is best to imagine the radiation of the source 
to have been stopped at a certain moment, so that there is a plane 
which we may call the rear of the progression of waves. It approaches 
the black disk with the velocity c, and if U and | S | have the same 
meaning as just now, the magnitude of the electromagnetic momentum 
will diminish by 

per unit of time. Consequently, there will be a normal pressure of 
this intensity acting on the disk. The result agrees with what we 
have deduced from the value of the stress in the ether, the quantity 
I S ] being related to the amplitude a by the equation 

It is easy to extend these results to a more general case. Let a 
plane disk receive, from any direction we like, a beam of parallel rays, 
and let one part of these be reflected, another absorbed and the 
remaining part transmitted. Let the vectors S, s' and s'' be the 
flows of energy per unit of area in the incident, reflected and trans- 
mitted beams, S, s', s" the mean flows taken for a full period, 
2, £', 2"- the normal sections of the beams. Then, if we imagine 
the space occupied by the light to be limited by two fronts, one in 
the reflected and one in the transmitted beam, and by a rear plane 
in the incident one, all these planes travelling onward with the 
velocity c, the change of electromagnetic momentum will be given 
by the vector expression 

and the force on the plate by 

^(2;s-2;'r-2:"s"). 

It must here be mentioned that the radiation pressure has been ob- 
served by Lebedew^) and by E. F. Nichols and HulP), and that 
the theoretical predictions as to its intensity have been verified to 
within one percent by the measurements of the last named physicists. 

1) P. Lebedew, Untersuchungen iiber die Druckkrafte des Lichtes, Ann. 
Phys. G (1901), p. 433. 

2) B. F. Nichols and G. P. Hull, The pressure due to radiation, 
Astrophysioal Journ. 17 (1903), p. 315; also Ann. Phys. 12 (1903), p. 225. 



FIELD or MOVING ELECTRONS. 35 

26. The theory of electromagnetic momentum, which we have 
found of so much use in the case of beams of light that are emitted, 
reflected or absorbed by a body, is also applicable to the widely 
different case of a moving electron. We may therefore, without too 
abrupt a transition, turn once more to some questions belonging to 
what we can call the dynamics of an electron, and in which we are 
concerned with the field the particle produces and the force exerted 
on it by the ether. We shall in this way be led to the important 
subject of the electromagnetic mass of the electrons. 

To begin with, I shall say some words about the field of a 
system of electrons or of charges disti-ibuted in any way, having a 
constant velocity of translation «', say in the direction of the axis 
of X, smaller than the speed of light c. We shall introduce axes of 
coordinates moving with the system, and we shall simplify our for- 
mulae by putting 

""=/3. (55) 



w 
e 



Now, we have already seen that the field is carried along by the 
system. The same may be said of the potentials cp and a, which 
serve to determine it, and it may easily be inferred from this^) that 

the values of ^ and -^ in a fixed point of space are given by 

dtp da. 

ox' ox 

Similarly 

dt^ ~^ dx^' dr- dx'- 

Thus the equation (31) takes the form 

whereas (32) may be replaced by the formula 

(1 - ^^) gf + ^^f + ^ = - ^9, (57) 

the components a^, and a^ being both 0, as is seen directly from (36). 
Comparing (56) and (57), we conclude that 

so that we have only to determine the scalar potential. 

This can be effected by a suitable change of independent 
variables. If a new variable x' is defined by 

x' = (l- p'y ''' X, (58) 

1) Note 13. 

3* 



36 I. GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 
(56) becomes „„ „„ „„ 

having the well known form of Poisson's equation. Since this 
equation occurs in the determination of the field for charges that 
are at rest, the problem is hereby reduced to an ordinary problem 
of electrostatics. Only, the value of ff in our moving system S is 
connected with the potential, not of the same system when at rest, 
but of a system in which all the coordinates parallel to OX have 
been changed in the ratio determined by (58).^) 

The result may be expressed as follows. Let 8' be a system 
having no translation, and which we obtain by enlarging the dimen- 
sions of 8 in the direction of X in the ratio of 1 to (1 — /3^) " . 
Then, if a point with the coordinates x, y, 2 in S and a point with 
the coordinates x', y, z in 8' are said to correspond to each other, 
if the charges of corresponding elements of volume are supposed to 
be equal, and if (f is the potential in S' , the scalar potential in the 
moving system is given by 

^ = (1 _ /32)- % y'. (60) 

Let us now take for the moving system a single electron, to which 
we shall ascribe the form of a sphere with radius jR and a uniformly 
distributed surface -charge e. The corresponding system 8' is an 
elongated ellipsoid of revolution, and its charge happens to be distri- 
buted according to the law that holds for a conductor of the same 
form. Therefore, the field of the moving spherical particle and all 
the quantities belonging to it, can be found by means of the ordinary 
theory of a charged ellipsoid that is given in many treatises. I shall 
only mention the results obtained for the more important quantities. 
The total electric energy is given by 

and the magnetic energy by 

As to the electromagnetic momentum, this has the direction of the 
translation, as may at once be deduced from (53), because we know 
already that the general direction of the flow of energy coincides 
with that of the motion of the particle. The formula for the magni- 
tude of the electromagnetic momentum, calculated for the first time 
by Abraham, is 

1) Note 14. 



FORCE ON KN ELECTRON, DUE TO ITS OWN FIELD. 37 

All these yalues U, T, { G | increase when the velocity is augmented. 
They become injfinite for /3 = 1, i. e. when the electron reaches a 
velocity equal to that of light. ^) 

27. According to our fundamental assumptions, each element of 
volume of an electron experiences a force due to the iield produced 
by the particle itself, and the question now arises whether there will 
be any resultant force acting on the electron as a whole. The con- 
sideration of the electromagnetic momentum will enable us to decide 
this question. 

If the velocity tv is constant in magnitude and direction, as it 
has been supposed to be in what precedes, the vector G will likewise 
be constant and there will be no resultant force. This is very im- 
portant; it shows that, if free from all external forces, an electron, 
just like a material point, will move with constant velocity, notwith- 
standing the presence of the surrounding ether. In all other cases 
however there is an action of the medium. 

It must be observed that, in the case of a variable velocity, the 
above formulae for U, T and | G | do not, strictly speaking, hold. 
However, if the variation of the state of motion is so slow that the 

change taking place in a time — may be neglected, one may apply the 

formula (63) for every moment, and use it to determine the change G 
of the momentum per unit of time.^) As the result depends on the 
acceleration of the electron, the force exerted by the ether is like- 
wise determined by the acceleration. 

Let us first take the case of a rectilinear translation with 
variable velocity w. The vector G is directed along the line of 
motion, and its magnitude is given by 

dt aw c dp 

Putting 

-J^- c d§ -^''' ^^^> 

we conclude that there is a force acting on the electron, opposite to 
its acceleration and equal to the product of the latter and the 
coefficient m. 

In the second place, I shall consider an electron having a 
velocity W of constant magnitude, but of varying direction. The 
acceleration is then normal to the path and it is convenient to use 
vector equations. Let W be the velocity, w the acceleration, and 



1) Note 15. 2) See S 37. 



38 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

let us take into account that in this case there is a constant i-atio 
between | G | and | W [ , for which I shall write 

LGJ = I G I = n"_ (65) 

j w I c/3 ^ ^ 

We have also 

G = m"\N, 

and the force exerted by the ether is given by 

— G = — m'^N . 

It is opposite in direction to the normal acceleration W and has an 
intensity equal to the product of this acceleration with the coeffi- 
cient m' . 

In the most general case the acceleration j will be directed 
neither along the path nor normally to it. If we decompose it into 
two components, the one j' in the line of motion and the other j" 
at right angles to it, we shall have, for the force on the electron 
due to its own electromagnetic field, in vector notation^) 

— vfi\ — m"\". (66) 

28. The way in which these formulae are usually interpreted 
will become clear to us, if we suppose the electron to have a certain 
mass «/g in the ordinary sense of the word, and to be acted on, not 
only by the force that is due to its own field, but also by a force K 
of any other kind. The total force being 

K - m'\ - m"\", 

the equation of motion, expressed in the language of vector analysis, 
will be 

K-m'j'-OT"j" = mo(j' + j"). (67) 

Instead of this we can write 

K = (nig -\- m')\' + (jwo + m")j", 

from which it appears that the electron moves, as if it had tivo different 
masses wig + m and m^ + m" , the first of which comes into play 
when we are concerned with an acceleration in the line of motion, 
and the second when we consider the normal acceleration. By 
measuring the force K and the accelerations j' and j" in different 
cases, we can determine both these coefficients. We shall call 
them the effective masses, m^ the material mass, and m' , m" the 
electromagnetic masses. In order to distinguish m' and m", we can 
apply the name of longitudinal electromagnetic mass to the first, and 

1) Note 16. 



ELECTROMAGNETIC MASS OF AN ELECTRON. 39 

that of transverse electromagnetic mass to the second of these 
coefficients. ■') 

From what has been said one finds the following formulae for 
m' and m": 

^" = i6^iW[- 2/3 + (1 + ^^) log l^], (69) 

or, expanded in series, 

For small velocities the two masses have the same value 

»»»' = '^" = 6«t ' C^2) 

whereas for larger velocities the longitudinal mass always surpasses 
the transverse one. Both increase with j3, until for /3 = 1, i. e. for 
a velocity equal to the speed of light, they become infinite. 

If, for a moment, we confine ourselves to a rectilinear motion 
of an electron, the notion of electromagnetic mass can be derived 
from that of electromagnetic energy. Indeed, this latter is larger for 
a moving electron than for one that is at rest. Therefore, if we are 
to put the particle in motion by an external force K, we must not 
only produce the ordinary kinetic energy -^m^w'^ but, in addition to 
this, the part of the electromagnetic energy that is due to the velo- 
city. The effect of the field will therefore be that a larger amount 
of work is required than if we had to do with an ordinary material 
particle mg-, it will be just the same as if the mass were larger 
than niQ. 

By reasoning of this kind we can also easily verify the for- 
mula (68). If the velocity is changing very slowly, we may at every 
instant apply the formulae (61) and (62). Since the total energy 
T -1" ?7 is a function of the velocity ic, its rate of change is given by 

'JI±!!>^ = 'Al±-Sll^. (73) 

This must be equal to the work done per unit of time by the moving 
force, or rather by the part of it that is required on account of the 



1) The notion of (longitudinal) electromagnetic masa was introduced for 
the first time by J. J. Thomson in his paper „0n the electric and magnetic 
effects produced by the motion of electrified bodies", Phil. Mag. (5) 11 (1881), 
p. 227. The result of his calculation is, however, somewhat different from that 
to which one is led in the modern theory of electrons. 



40 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

electromagnetic field. Consequently, dividing (73) by -w, we find the 
intensity of this part, and if next we divide by w, the acceleration, 
the result must be the longitudinal electromagnetic mass. If one 

calculates 

,__ 1 d{T+TD _ 1 d{T^TJ) 
cw d§ /?c^ dp 

by means of the formulae (61) and (62), one really finds exactly the 
value (68). 

29. A close analogy to this question of electromagnetic mass 
is furnished by a simple hydrodynamical problem. A solid, per- 
fectly smooth sphere, moving with the velocity W in an incom- 
pressible perfect fluid which extends on all sides to infinite distance, 
produces in this fluid a state of motion characterized by a kinetic 
energy for which we may write 

if a is a constant, depending on the radius of the ball and on the 
density of the fluid. Under the influence of an external force applied 
to the ball in the direction of the translation, its velocity will change 
as if it had, not only its true mass m^, but besides this an apparent 
mass m', whose value is given by 

1 dT 

w| d I w I ' 

a formula corresponding to the last equation of § 28. 

We could have obtained the same result if we had first calculated 
the momentum of the fluid. We should have found for it 

G = aW, 

an expression from which we can also infer that the transverse 
apparent mass has the same value « as the longitudinal one. 
This is shown by the equation 

., |G| 
m = — \ ■ 

[w| 

30. If, in the case of the ball moving in the perfect fluid, we 
were obliged to confine ourselves to experiments in which we measure 
the external forces applied to the body and the accelerations produced 
by them, we should be able to determine the effective mass m^ -\- m 
(or m^ + m"), but it would be impossible to find the values of m^ 
and m (or m") separately. Now, it is very important that, in the 
experimental investigation of the motion of an electron, we can go 
a step farther. This is due to the fact that the electromagnetic mass 
is not a constant, but increases with the velocity. 



RATIO OF THE CHARGE TO THE MASS OF AN ELECTRON. 41 

Suppose we can make experiments for two different known 
velocities of an electron, and that by this means we can find the 
ratio Tc between the effective transverse masses which come into play 
in the two cases. Let ^ be the ratio between the electromagnetic 
transverse masses, calculated, as can really be done, by the formula (69). 
Then, distinguishing by the indices I and II the quantities relating 
to the two cases, we shall have the formulae 

"'o + m __ mj _ 

+ /f ''^y ft ^j 

mn mil 

and the ratio between the true mass m^ and the electromagnetic 
mass m'j will be given by 

»l/ _ X (fc — 1) 

If the experimental ratio li differed very little from the ratio % that 
is given by the formula (69), m^ would come out much smaller 
than mi and we should even have to put m^ = 0, if /c were exactly 
equal to %. 

I have spoken here of the transverse electromagnetic mass, because 
this is the one with which we are concerned in the experiments I 
shall now have to mention. 

31. You all know that the cathode rays and the /3-rays of 
radio-active bodies are streams of negative electrons, and that Gold- 
stein's canal rays and the w-rays consist of similar streams of 
positively charged particles. In all these cases it has been found 
possible to determine the ratio between the numerical values of the 
charge of a particle and its transverse effective mass. The chief 
method by which this has been achieved is based on the measure- 
ment, for the same kind of rays, of the deflections from their recti- 
linear course that are produced by known external electric and 
magnetic forces. 

The theory of the method is very simple. If, in the first place, 
an electron having a charge e and an effective mass m, moves in an 
electric field d, with a velocity w perpendicular to the lines of force, 

ed 

the acceleration is given by — ; hence, if r is the radius of curvature 

of the path, 

w^ e I d I 

r m ' 

so that, if I d I and r have been measured, we can calculate the 
value of 

~r (74) 



42 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

Let us consider in the second place an electron moving in a magnetic 
iield h, and let us suppose the velocity tv to be perpendicular to the 

magnetic force. Then, the field will exert on the particle a force — ^-^ , 
as is seen from the last term of (23). This force being perpendicular 
to the velocity, we shall have, writing / for the radius of curvature 
of the path, 

M)' e4« [ h I 
r cm 

The determination of | h | and r' can therefore lead to a knowledge 
of the expression 



and, by combining this with (74), we shall be enabled to find both w 

and — • 
m 

32. I shall not speak of the large number of determinations of 
this kind that have been made by several pihysicists, and will only say 
a few words relative to the important work of Kaufmann-*^) on the j3- 
rays of radium. These rays appear to contain negative electrons with 
widely different velocities, so that it is possible to examine the question 

whether — is a function of the velocity or a constant. Kaufmann's 

experiments were arranged in such a manner that the electric and 
the magnetic deflection, belonging to the saine electrons, could be 

measured, so that the values both of iv and of ~ could be deduced 

' HI 

from them. Now, it was found that, while the velocity iv ranges from 

about 0,5 to more than 0,9 of the velocity of light, the value of — 

diminishes considerably. If we suppose the charge to be equal for 

all the negative electrons constituting the rays, this diminution of — 

must be due to an increase of the mass tn. This proves that at all 
events the electromagnetic mass has an appreciable influence. It must 
even greatly predominate. Indeed, Kaufmann's numbers show no 
trace of an influence of the material mass Wg, his ratio 7c of effective 
masses for two different velocities (a ratio which is the inverse of 

that of the values of -H agreeing within the limits of experimental 

errors with the ratio x between the electromagnetic masses, as deduced 
from Abraham's formula (69). 



1) W. Kaufmann, tJber die Konstitution des Elektrons, Ann. Phya. 19 
(1906), p. 487, 



MASS OF A NEGATIVE ELECTRON WHOLLY ELECTROMAGNETIC. 43 

Of course, we are free to believe, if we like, that there is some 
small material mass attached to the electron, say equal to one 
hundredth part of the electromagnetic one, but with a view to 
simplicity, it will be best to admit Kaufmann's conclusion, or hypo- 
thesis, if we prefer so to call it, that the negative electrons have no 
material mass at all. 

This is certainly one of the most important results of modern 
physics, and I may therefore be allowed to dwell upon it for a short 
time and to mention two other ways in which it can be expressed. 
We may say that, in the case of a moving negative electron, there 
is no energy of the ordinary form \mf^tv^, but merely the electro- 
magnetic energy T -\- U, which may be calculated by means of the 
formulae (61) and (62). For high velocities this energy is a rather 
complicated function of the velocity, and it is only for velocities 
very small compared with that of light, that the part of it which 
depends on the motion, can be represented by the expression -l-ni'tv^, 
where m' has the value given by (72). This is found by expanding 
T -\- U in a, series similar to (70) and (71). 

We obtain another remarkable form of our result, if in the 
equation of motion (67), which for wJq = becomes 

Kf If If \ff A 

— m \ ~in \ =0, 

we attach to the two last terms their original meaning of forces 
exerted by the ether. The equation tells us that the total force 
acting on the particle is always 0. An electron, for example, which 
has an initial velocity in an external electromagnetic field, will move 
in such a manner that the force due to the external field is exactly 
counterbalanced by the force that is called forth by the electron's 
own field, or, what amounts to the same thing, that the force exerted 
by the resulting field is 0. 

After all, by our negation of the existence of material mass, the 
negative electron has lost much of its substantiality. We must make 
it preserve just so much of it, that we can speak of forces acting 
on its parts, and that we can consider it as maintaining its form and 
magnitude. This must be regarded as an inherent property, in virtue 
of which the parts of the electron cannot be torn asunder by the 
electric forces acting on them (or by their mutual repulsions, as we 
may say). 

33. In our preceding reasoning we have admitted the equality 
of the charges of all the negative electrons given off by the radium 
salt that has been used in Kaufmann's experiments. We shall now 
pass on to a wide generalization of this hypothesis. 



4-1 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

As is well known, Faraday's law of electrolysis proves that all 
monovalent electrolytic ions have exactly equal charges, and that, 
if this is denoted by e, the charges of bivalent, trivalent ions etc. 
are 2e, 3e etc. Thus the conception has arisen that this e, say the 
charge of an ion of hydrogen, is the smallest quantity of electricity 
that ever occurs in physical phenomena, an atom of electricity, as 
we may call it, which can only present itself in whole numbers. 
Experimental determinations by J. J. Thomson^) of the charges 
carried by the ions in conducting gases, and certain speculations about 
the electrons which are vibrating in a body traversed by a beam of 
light, have made it highly probable that this same amount of charge e 
occurs in these cases, that it is, so to say, a real natural unit of 
electricity, and that all charged particles, all electrons and ions carry 
one such unit or a multiple of it. The negative electrons which 
constitute the |3-rays and the cathode rays are undoubtedly the 
simplest of all these charged particles, and there are good reasons for 
supposing their charge to be equal to one unit of electricity, i. e. to 
the charge of an ion of hydrogen. 

Leaving aside the case of multiple charges, and ascribing to all 
electrons or ions, whether they be positive or negative, the same 
amount of electricity, we can say that the masses m of different 
particles are inversely proportional to the values that have been found 

for — • 
m 

Now, for the negative electrons of the cathode rays and of the 

j3-rays, this latter value is, for small velocities^) 

1,8- louyi^. 

For an ion of hydrogen, the corresponding number can be drawn 
from the electrochemical equivalent of the gas. It is found to be 

nearly 1900 times smaller than the number for the free negative 
electrons. Hence, the mass of a negative electron is about the 1900''' 
part of that of an atom of hydrogen. 



1) See J. J. Thomson, Conduction of electricity through gases, and The 
corpuscular theory of matter, London, 1907, by the same author. 

2) I write it in this form in order to show that the number is 1,8 ■ 10', 
if the ordinary electromagnetic units are used. It may be mentioned here that 
Simon's measurements on cathode rays [Ann. Phys. Ghem. C9 (1899), p. 589] 
lead to the value 1,878-10', and that Kaufmann, calculating his results by 
means of Abraham's formulae, finds 1,823-10'. Later experiments on /3-rays 
by Bestelmeyer [Ann. Phys. 2'2 (1907), p. 429], however, have given the 
number 1,72 - 10'- 



ELECTROMAGNETIC THEORY OF MATTER. 45 



,? It must be noticed especially that the values of - obtained for 
.-r — ~-~^_^ J- ./ „j 

(different^ negative electrons are approximately equal. This lends a 

strong support to the view that all negative electrons are equal to 

each other. On the contrary, there are great differences between the 

positive electrons, such as we find in the canal rays and the a-rays 

of radio-active substances. The values of ~ belonging to these rays 

are widely divergent. They are however all of the same order of 
magnitude as the values holding for electrolytic ions. Consequently, 
the masses of the positive electrons must be comparable with those of 
chemical atoms. We can therefore imagine the free electrons to be 
the product of a disintegration of atoms, of a division into a positively 
and a negatively charged particle, the first having nearly the whole 
mass of the atom, and the second only a very small part of it. 

34. Of late the question has been much discussed, as to whether 
the idea that there is no material but only electromagnetic mass, which, 
in the case of negative electrons, is so strongly supported by Kauf- 
mann's results, may not be extended to positive electrons and to 
matter in general. On this subject of an electromagnetic theory of 
matter we might observe that, if we suppose atoms to contain negative 
electrons, of which one or more may be given off under certain 
circumstances, as they undoubtedly are, and if the part that remains 
after the loss of a negative particle is called a positive electron, then 
certainly all matter may be said to be made up of electrons. But 
this would be mere words. What we really want to know is, 
whether the mass of the positive electron can be calculated from the 
distribution of its charge in the same way as we can determine the 
mass of a negative particle. This remains, I believe, an open question, 
about which we shall do well to speak with some reserve. 

In a more general sense, I for one should be quite willing to 
adopt an electromagnetic theory of matter and of the forces between 
material particles. As regards matter, many arguments point to the 
conclusion that its ultimate particles always carry electric charges 
and that these are not merely accessory but very essential. We 
should introduce what seems to me an unnecessary dualism, if we 
considered these charges and what else there may be in the particles 
as wholly distinct from each other. 

On the other hand, I believe every physicist feels inclined to the 
view that all the forces exerted by one particle on another, all 
molecular actions and gi-avity itself, are transmitted in some way by 
the ether, so that the tension of a stretched rope and the elasticity 
of an iron bar must find their explanation in what goes on in the 



46 I- GENERAL PRINCIPLES. THEORY OP FREE ELECTRONS. 

ether between the molecules. Therefore, since we can hardly admit 
that one and the same medium is capable of transmitting two or 
more actions by wholly different mechanisms, all forces may be re- 
garded as connected more or less intimately with those which we 
study in electromagnetism. 

For the present, however, the nature of this connexion is entirely 
unknown to us and we must continue to speak of many kinds of 
forces without in the least being able to account for their origin. 
We shall even be obliged to subject the negative electrons to certain 
forces, about whose mode of action we are in the dark. Such are, for 
example, the forces by which the electrons in a ponderable dielectric 
are driven back to their positions of equilibrium, and the forces that 
come into play when an electron moving in a piece of metal has 
its course changed by an impact against a metallic atom. 

35. The universal unit of electricity of which we have spoken 
can be evaluated as soon as we have formed an estimate of the 
mass of the chemical atoms. This has been done fairly well in 
different ways, and we shall not be far from the truth if we take 

1,3 -10-^^ gramm 
for the mass of an atom of hydrogen. Combining this with the 
electrochemical equivalent of this element, which in our units is 

— — , we find for the charge of an ion of hydrogen 

cyiTt 

l,3-10-2°cl/i^. 
This number must also represent the charge of a negative electron. 
Therefore, the value of — (for small velocities) being 

1,8 lO'cl/i^, 
we find 

Mi = 7 • 10~^* gramm. 

Now, this must be the mass given by the formula (72). Substituting 
also the value of e, we get the following number for the radius 

E= 1,5-10-" cm. 

We may compare this with the estimates that have been formed in 
the kinetic theory of gases. The distance of neighboring molecules 
in the atmospheric air is probably about 



and the diameter of a molecule of hydrogen may be taken to be 

2-10-8 cm. 



ELECTROMAGNETIC MASS OF A SYSTEM OF ELECTRONS. 47 

You see that, compared with these lengths, the electron is quite 
microscopical. Probably it is even much smaller than a single atom, 
so that, if this contains a certain number of negative electrons, these 
may be likened to spheres placed at distances from each other that 
are high multiples of their diameters. 

36. Before closing our discussion on the subject of electro- 
magnetic mass, I must call your attention to the question as to whether, 
in a system composed of a certain number of electrons, tbe electro- 
magnetic mass is the sum of the electromagnetic masses of the 
separate particles, or, as I shall rather put it, whether, if the system 
moves with a common velocity of translation, the electromagnetic 
energy, in so far as it depends on the motion, can be made up of parts, 
each belonging to one electron, so that, for small velocities, it can 
be represented by 

This will, of course, be the case, if the electrons are so far apart that 
their fields may be said not to overlap. If, however, two electrons 
were brought into immediate contact, the total energy could not 
be found by an addition, for the simple reason, that, being a 
quadratic function of d and h, the energy due to the superposition 
of two fields is not equal to the sum of the energies which would 
be present in each of the two, if it existed by itself. 

We have now to bethink ourselves of the extreme smallness of 
the electrons. It is clear that the larger part of the electromagnetic 
energy belonging to a particle will be found in a very small part of 
the field lying quite near it, within a distance from the centre that 
is a moderate multiple of the radiusi , Therefore, it may very well be 
that a number of electrons are so widely dispersed that the effective 
parts of their fields lie completely outside each other. In such a 
case the system may be said to have an electromagnetic mass equal 
to the sum of the masses of the individual electrons. 

Yet there are important cases in which we are not warranted in 
asserting this. In order to make this clear, I shall call F^ the part of 
the field of an electron which lies nearest to the particle, and F^ the 
more distant part, the surface of separation between the two being a 
sphere whose radius is rather large in comparison with that of the 
electron. Then, if the electron is taken by itself, the part E^ of the 
energy contained within F^ far surpasses the energy E^ which has its 
seat in F,^. Now, if we have N electrons at such distances from each 
other that their fields F^ do not overlap, we shall have to add to 
each other the amounts of energy E^. The quantities E.^ on the 
contrary must not be simply added, for the remoter fields F^ will 
certainly cover, partly at least, the same space S. If, in this space, 



48 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

the dielectric displacements or the magnetic forces due to the indi- 
vidual eleitrons have directions making rather small angles with each 
other, all the fields F^, feeble though they are, may very well pro- 
duce a resulting field of appreciable energy. We have an example of 
this in the electric field of a charged conductor, and in the magnetic 
field aroimd a wire carrying a current. The energy of this magnetic 
field may be shown to be, in very common cases, considerably larger 
than the sum of all the amounts of energy which I have called E^, 
at least in as much as these depend on the motion of the electrons. 
The possibility of this will be readily understood, if one thinks of 
the extreme case that, at a point of the space S, all electrons produce 
a magnetic force in exactly the same direction. Then, if each of these 
forces has the magnitude j h | , the resultant magnetic force has the 
magnitude iV | h ] , so that the magnetic energy per unit of volume 
becomes \ N^ h^. This is proportional to the square of the number N 
which we shall suppose to be very large. On the other hand, the 
sum of the quantities E^ may be reckoned to be proportional to the 
first power of N. 

This digression was necessary in order to point out the connexion 
between the electromagnetic mass of electrons and the phenomena of 
self-induction. In these latter it is the magnetic energy due to the 
overlapping of the feeble fields F^ that makes itself felt. In dealing 
with effects of induction we can very well speak of the electro- 
magnetic inertia of the current, or of the electromagnetic mass of the 
electrons moving in it, but we must keep in mind that this mass is 
very much larger than the sum of those we should associate with 
the separate particles. This large value is brought about (as are all 
effects of the current) by the cooperation of an immense number of 
electrons of the same kind moving in the same direction."^) 

37. In our treatment of the electromagnetic mass of electrons 
we have started from the expression (66) for the force to which an 
electron is subjected on account of its own field. However, this ex- 
pression is not quite exact. It is based on the assumption that the 
equation (63) may be applied to a case of non-uniform motion, and 
we observed already that this may be done only if the state of motion 
changes very little during the time an electromagnetic disturbance 
would take to travel over a distance equal to the dimensions of the 
electron. This amounts to saying that, if I is one of these dimen- 
sions, and T a time during which the state of motion is sensibly 
altered, the quantity 

i (75) 

must be very small. 

1) Note 17. 



RESISTANCE TO THE MOTION. 49 

In reality, the force (66) is only the first term of a series in 
which, compared with the preceding one, each term is of the order 
of magnitude (75). 

In some phenomena the next term of the series makes itself felt; 
it is therefore necessary to indicate its value. By a somewhat la- 
borious calculation it is found to be 



67rc» 



(76) 

where the vector V is twice differentiated with respect to the time. 
I may mention by the way that this formula holds for any distri- 
bution of the electric charge e}) 

In many cases the new force represented by (76) may be termed 
a resistance to the motion. This is seen, if we calculate the work of 
the force during an interval of time extending from t = t.^ to t = t^. 
The result is 

Here the first term disappears if, in the case of a periodic motion, 
the integration is extended to a full period; also, if at the instants t^ 
and t^ either the velocity or the acceleration is 0. We have an 
example of the latter case in those phenomena in which an electron 
strikes against a ponderable body and is thrown back by it. 

Whenever the above formula reduces to the last term, the work 
of the force is seen to be negative, so that the name of resistance is 
then justly applied. This is also confirmed by the form our formula 
takes for an electron having a simple harmonic motion. The velocity 
being given by 

V = bcosM^, 

where w is a constant, we may write V = — w°V, and, instead of (j&), 

— -a — 3-v, (77) 

so that, in this particular case, the force is opposite to the velocity 
and proportional to it. 

The work of (77) during a full period T is 

- -f 'v f^'dt = - -^ b'T. (78) 



38. In all cases in which the work of the force (76) is nega- 
tive, the energy of the electron (if not kept at a constant value by 



1) Note 18. 

Lorentz, Theory of electrons. 



50 I- GENERAL PEINCIPLES. THEORY OP FREE ELECTRONS. 

the action of some other cause) must diminish, and that of the ether 
must increase. This means that there is a continous radiation from 
the particle dutwards, such as cannot be said to exist when the velo- 
city is constant and the electron simply carries its field along with it. 

For the purpose of getting a clear idea of the radiation, it is 
well to consider the field at a very large distance from the particle. 
We shall see that, if the distance is large enough, the radiation field 
gets, so to say, disentangled from the field we have formerly consi- 
dered, which is carried along by the moving particle. 

In order to determine the field at a large distance, we can avail 
ourselves of the following formulae for the scalar and the vector 
potential, which hold for all points whose distance from the electron 
is very large compared with its dimensions: 

„ t , a ^w m) 

Here, the square brackets have a meaning similar to that which we 
gave them in the general equations (35) and (36). If one wishes to 
determine the potentials at a point P for the time t, one must first 
seek the position M of the electron, which satisfies the condition 
that, if it is reached at the time t^, previous to t, 

3IP^c(t-Q. 

The distance il/P is denoted by r, and [v] means the velocity in the 
position M, V^ its component in the direction il/P. 

The formulae have been deduced from (35) and (36)-, the vector 

1 '- in the denominators shows, however, that the problem is not 

quite so simple as might be expected at first sight. A complication 
arises from the circumstance that we must not integrate over the 
space occupied by the electron at tlie particular instant icliich we 
have denoted hy t^. On the contrary, according to the meaning 
of (35) and (3G), we must fix our attention on the diiferent points 
of the electron and choose for each of them, among all its succes- 
sive positions, the one 31' which is determined by the condition, 
that, if it is reached at the time to, 

M' P == c{t - ti) . 

The time t'^ is slightly different for the different points of the electron 
and therefore the space over which we have to integrate (which 
contains all the points il/') cannot be said to coincide with the 
space occupied by the electron at any particular instant.^) 

1) Note 1!). 



RADIATION FROM AN ELECTRON. 51 

39. Leaving aside these rather complicated calculations, I pro- 
ceed to the determination of the field at very large distances. The 
formulae (33) and (34) which we must use for this purpose require 
us to diiferentiate cp and a. In doing so I shall omit all terms in 
which the square and the higher powers of the distance r appear in 
the denominator. I shall therefore treat as a constant the factor r in 
the denominators of (79), so that only V^ has to be differentiated in the 
expression for qp and, if we also neglect terms in which a component 
of the velocity is multiplied by one of the acceleration, only [v] in 
the second formula. Performing all operations and denoting by x, y, z 
the coordinates of P with respect to the point M as origin, and by j 
the acceleration of the electron in the position M, I find^) 



d = 



47rc*r 



■(-L + yJ.), etc. (80) 



z 



in c^r \^y r ■'= r 



\y~-hf ,^t<^- (81) 



The three formulae for tl can be interpreted as follows. If the acce- 
leration j is decomposed into j^ in the direction of IIP and j per- 
pendicular to it, the dielectric displacement in P is parallel to j and 
its magnitude is given by 



iiic^r 



In order to see the meaning of the equations for h, we can introduce 
a vector k of unit length in the direction from 31 towards P. The 

components of this vector being — , — , — , we have 



'.,[j-l<l- 



The magnitude of h is therefore 

e 

equal to that of the dielectric displacement. Further, the magnetic 
force is seen to be perpendicular both to the line MP and to the 
dielectric displacement. Consequently there is a flow of energy along 
MP. It is easily seen that this flow is directed away from the 
position M of the electron, and that its intensity is given by 

if d- is the angle between il/P and the acceleration j.^) 



1) Note 20. 2) Note 21. 



52 I. GENERAL PRINCIPLES. THEORY OP PBEE ELECTRONS. 

The result may be applied to any point of a spherical surface 
described around the centre M with r as radius. The total outward 
flow of energy across this sphere is given by 



ISTt'c^r^ ' J ewe" ■■ ^ ^ 

The reason for my former assertion that, at very large distances 
from the electron, the radiation field predominates over the field con- 
sidered in § 26, lies in the fact that, in the latter, d and h diminish 

as —J- and in the radiation field only as -;- • 

We can sum up the preceding considerations by saying that an 
electron does not emit energy so long as it has a uniform rectilinear 
motion, but that it does as soon as its velocity changes either in 
magnitude or in direction. 

40. The theory of the production of Rontgen rays, first pro- 
posed by Wiechert and Stokes, and worked out by J. J. Thomson^), 
affords a very interesting application of our result. According to it, 
these rays consist of a rapid and iiTegular succession of sharp electro- 
magnetic impulses, each of which is due to the change of velocity 
which an electron of the cathode rays undergoes when it impinges 
against the anti-cathode. I cannot however dwell upon this subject, 
having too much to say about the emission of light -vibrations with 
which we shall be often concerned. 

If an electron has a simple harmonic motion, the velocity is 
continually changing, and, by what has been said, there must be a 
continous emission of energy. It will also be clear that, at each point 
of the surrounding field, the state is periodically changing, keeping 
time with the electron itself, so that we shall have a radiation of 
homogeneous light. Before going into some further details, I shall 
first consider the total amount of energy emitted during a full period. 

Let us choose the position of equilibrium as origin of coordinates 
and let the vibration take place along the axis of x, the displacement 
at the time t being given by 

X = a cos (nt -{- p). 

Then the acceleration is 

— an^ cos (nt + 2'')- 

1) E. Wiechert, Die Theorie der Blektrodynamik und die Rontgen'sche 
Entdeckung, Abh. d. Phys.-okon. Ges. zu Konigsberg i. Pr. (1896), p. 1; Ober die 
Grundlagen der Elektrodynamik, Ann. Phya. Cliem. 59 (1896),]). 283; 6. G. Stokes 
On the nature of the Rontgen rays, Manoh. Memoirs 41 (1897), Mem. 15- 
J. J. Thomson, A theory of the connexion between cathode and Ronto-en rays' 
Phil. Mag. (5) 46 (1898), p. 172. ° ' 



FIELD OP A VIBRATING ELECTRON. 53 

If the amplitude a is very small, the sphere of which we have npoken 
in the preceding paragraph may be considered as having its centre, 
not in Mj one of the positions of the electron, but in the origin 0, 
and we may understand by j the acceleration of the electron at the 

time t , r being the distance from 0. Therefore, on account 

of (82), the flow of energy across the sphere will be per unit of time 

Integrating this over a full period T we get 

--/' ,, a'n'T. (S3) 

Now, if the amplitude is to remain constant, the electron must be 
acted on by an external force equal and opposite to the resistance (77). 
The work of this force is given by (78) with the sign reversed. 
Since the amplitude of the velocity is equal to the amplitude a of 
the elongation, multiplied by n, the work of the force corresponds 
exactly to the amount of energy (83) that is emitted.^) 

41. For the sake of further examining the field produced 
by an electron having a simple harmonic motion, we shall go back 
to the formulae (79). Let us first only suppose that the motion 
of the electron is confined to a certain very small space S, one 
point of which is chosen as origin of coordinates. Let x, y, z 
be the coordinates of the electron, x, y, z its velocities and x, y, z 
the components of its acceleration. We shall consider all these quan- 
tities as infinitely small of the first order, and neglect all terms 
containing the product of any two of them. We shall further denote 
by X, y, z the coordinates of the point P for which we wish to 
determine the field, and by r^ its distance from the origin. Now, 
if M is the position of the electron of which we have spoken in our 
explanation of the equations (79), the distance il/P = r will be in- 
finitely near the distance r^, and the time i^ infinitely near the time 

t — ^-- The changes in the position and the velocity of the electron 

in an infinitely small time being quantities of the second order, we 

may therefore understand by M the position at the instant t 

and by V the velocity at that time. Further: 

I) Note 22. 



^0 



54 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

because, as is easily seen, the change in the distance between and P, 
due to a shifting of the first point towards M, is equal to the change 
that would take place, if remained where it was, but P were given 
a displacement — x, — y, — z. The square brackets now serve to in- 
dicate the values at the time if — —; they will have this meaning in 
all formulae that are now to be developed. 
Substituting the above value of — , and 

1 = 1 + .[^'■]- 



l__[Vrl ^ ' c ' 
c 

where V^ may be considered as the component along OP, we find 

for the scalar potential 

HavLUg got thus far, we can omit the index o, so that r now means 
the distance from the origin to a point with the coordinates x, y, z. 
As regards the last term, we can use the transformation 

a[x] ^ a[y] , a[z]^ 



/8[x] g[yj aLzj\ 



the last step in which will be clear, if we attend to the meaning 

of -^-- etc. The symbol [x] represents the value of x at the time i — — 

which we shall, for a moment, denote by H . This time f depends in 
its turn on the distance r, which again is a function of the coordi- 
nates X, y, 3 of the external point. Hence 

aw^^a*:|^^ 1. etc. 

dx dt dr dx '- -' c r ' 

Finally, the scalar potential becomes 

^= « |i_fM_f M_f Ml. (84) 

^ i7t [ r ox r ay r oe r ] ^ ■' 

The expression for the vector potential is even more simple, viz. 

a = -^. (85) 

The radiation field, which predominates at large distances, and in 
which we find the flow of energy of which we have already spoken, 
is determined by the three last terms of qp and by the vector potential. 
At smaller distances it is superposed on the field represented by the 
first term of (p, which is the same that would surround the electron 
if it were at rest. 



FIELD OF A POLAlilZBD PARTICLE. 55 

42. By a slight change in the circumstances of the case, we 
can do away with the electrostatic field altogether. Let us suppose 
the electron to perform its vibrations in the interior of an atom or a 
molecule of matter, to which we shall now give the name of particle 
and which occupies the small space S. If the particle as a whole 
is not charged, it must contain, besides our movable electron, a 
charge — e, either in the form of one or more electrons, or distri- 
buted in any other manner. We shall suppose that this complementary 
charge — e remains at rest, and that, if the electron e did so likewise, 
in a determinate position, which we shall take as origin of coordinates, 
there would be no external field at all, at least not at a distance that 
is large in comparison with the dimensions of S. This being admitted, 
the immovable charge — e must produce a scalar potential equal and 
opposite to the first term in (84), so that, if we consider the field 
of the whole particle, this term will be cancelled. Our assumption 
amounts to this, that the charge — e is equivalent to a single electron — e 
at the point 0, so that, if the electron + e has the coordinates x, y, z, 
things will be as if we had two equal and opposite charges at a 
small distance from each other. We express this by saying that the 
particle is electrically polarized, and we define its electric moment by 

the equation 

p = 6r, (86) 

where r is the vector drawn from towards the position of the 
movable electron. The components of p are 

Pa: = ex, p, = ey, p, = ez, (87) 

and from (84) and (8b) we find the following expressions for the 
potentials in the field surrounding the polarized particle 

1 f 3 [PJ , ^ M , 3 [PJ 1 f88) 



[P] 
incr 



(89) 

These relations also hold in the case of a polarized particle whose 
state is somewhat more complicated. Let us imagine that it contains 
a certain number of electrons, any part of which may be movable. 
We shall find the potentials by calculating (84) and (85) for the 
separate electrons and adding the results. Using the sign 2J for this 
last operation, and keeping in mind that 

26 = 0, (90) 

we shall again find the formulae (88) and (89), if we define the 
moment of the particle by the formula 

p = ^er, (91) 



pz 



dS, 



(93) 



(94) 



56 I. GENERAL PKINCIPLES. THEORY OF FREE ELECTRONS. 

or its components by 

p,= 2:ex, p,, = 2;ey, p, = I^ez. (92) 

It is even unnecessary that the charges should be concentrated in 
separate electrons. We can as well suppose them to be continuously 
distributed, but of course capable of moving or fluctuating in one 
■way or another. Then the sums in the last formulae must be replaced 
by integrals. We shall have 

and for the components of the moment 

P.-fQ^<iS, ^,j=fQrd8, p, =/( 

the integration being extended over the space S occupied by the 
particle. It must be noticed that on account of (90) and (93), the 
vectors (91) and (!'4) are independent of the choice of the point 0. 

43. The formulae (88) and (89) show that the particle is a 
centre of radiation whenever the moment p is changing, and that it 
emits regular vibrations if p is a periodic function of the time. 

We shall suppose for example that 

P, = 6cos(m^ + p), p^ = 0, p, = 0, 
h, n and p being constants. Then we have 

and the field is easily determined by means of (88) and (89). 

I shall not write down the general formulae but only those 
which hold for values of r that are very large compared with the 
wave-length, and which are obtained by the omission of all terms of 

They are as follows: 

n^h r^ — x^ f /, r\ i 



the order — r 



d. = 

h =0, 






■ cos n 



ocy 
xz 



cos\^n(t- ~)+p 



cos \n[t 



n^b 



z 



h. = 



n^b y 
r 



cos m(^ 



corresponding to (80) and (81).^) 
1) Note 23. 



cos \^n(t - 'j^ + p 



(95) 



FUNDAMENTAL EQUATIONS FOR A MOVING SYSTEM. 57 

I must add that our formulae for the field around a particle 
whose state of polarization is periodically changing, agree with 
those by which Hertz represented the state of the field around his 
vibrator.'^) 

44. We shall now pass on to certain equations that will 
be of use to us when we come to speak of the influence of the 
Earth's translation on optical phenomena. They relate to the electro- 
magnetic phenomena in a system of bodies having a common uniform 
translation, whose velocity we shall denote by W, and are derived 
from our original equations by a change of variables. Indeed, it is 
very natural to refer the phenomena in a moving system, not to a 
system of axes of coordinates that is at rest, but to one that is fixed 
to the system and shares its translation; these new coordinates will 
be represented by x' , y , z . They are given by 

X' = X-\NJ, y' = ij-\Hj, z' = Z-\Nj. (96) 

It will also be found useful to fix our attention on the velocity U 
of the charges relatively to the moving axes, so that in our funda- 
mental equations we have to put 

V = w + u. 

Now it has been found that in those cases in which the velocity of 

translation W is so small that its square W^, or rather the fraction -^, 

may he neglected, the differential equations referred to the moving 
axes take almost the same form as the original formulae, if, instead 
of t, we introduce a new independent variable t', and if, at the same 
time, the dielectric displacement and the magnetic force are replaced 
by certain other vectors which we shall call d' and h'. 
The variable t' is defined by the equation 

t' = t-^ (w^x + v/,/ + V//1 (97) 

and the vectors d' and h' by 

d' = d-f-l[w.h], (98) 

h' = h-i[w.d]. (99) 

We can regard t' as the time reckoned from the instant 



1) H. Hertz, Die Krafte elektrischer Schwingungen , behandelt naoli der 
Maxwell'schen Theorie, Ann. Phya. Chem. 36 (1888), p. 1. 



58 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

which changes from one point to the other. This variable is there- 
fore properly called the local time, in order to distinguish it from 
the universal time t. 

As to the vectors d' and h', the difference between them and d, 

h is but small, since the fraction —~ is so. Even if we have to do 
with the translation of the Earth, the value of | W j is no more than 
one ten -thousandth part of the velocity of light. 

Neglecting terms with the square of --- , as has already been 
said, one finds the following system of transformed equations: 

divd'==|l-(^),, (100) 

div h' = 0, (101) 

roth'= J((i' + 9U), (102) 

rotd'=-^h'. (103) 

The dot means a differentiation with respect to t', and the 
symbols div and rot (and, in the next paragraph, grad) serve to indicate 
differentiations with respect to x, y', e' in exactly the same manner 
as they formerly indicated differentiation,s with respect to x, y, s. 
Rot h', for example, now means a vector whose components are 

0h^ 3 hi/ Shi 3 hi ShJ, 3 hi 

dy dz dz dx' dx dy 

You see that the formulae have nearly, but not quite, the same form 

as (17) — (20), the difference consisting in the term -^— in the first 

equation.-') 



c 



45. Starting from the new system of equations, we can now 
repeat much of what has been said in connexion with the original 
one. For a given distribution and motion of the charges, the field 
is entirely determined, and here again the problem can be considerably 
simplified by the introduction of two potentials, a scalar and a vector 
one. These are given by the equations 



and 



where however the symbols [p] and [pu] require some explanation. 
If we want to calculate- qo' and a' for a point P, for the moment at 

1) Note 24. 



RADIATION FROM A MOVING PARTICLE. 59 

which the local time of this point has a definite value t', we must, 
for each element dS situated at a distance r from F, take the values 
of Q and Qti such as they are at the instant at which the local time 
of the element is 

c 
Finally, we have the following formulae for the determination 
of the field by means of the potentials^): 

d' = - i a' - grad <p' + j grad (w • a'), (106) 

h' = rot a'. (107) 

Here again, if we compare with (33) and (34), we notice a slight 
difference. In (33) there is no term corresponding to the last one 
in (106). 

Notwithstanding the two differences I have pointed out, there 
is a large variety of cases, in which a state of things in a system 
at rest has its exact analogue in the same system with a translation. 
I shall give two examples that are of interest. 

In the first place, the values of d' and h' produced by a jDarticle 
moving with the velocity W, and having a variable electric moment, 
are given by formulae similar to those we formerly found for the 
radiation from a particle without a translation, und which I therefore 
need not even write down. 

If the moment of a particle placed at the origin of coordinates 
is represented by 

p, = h cos (nt'+p), p^ = 0, p, = 0, (108) 

all we have to do is to replace, in (95), d, h, x, y, z, t by d', h', x, y, z , t'}) 

In order to show the meaning of this result, I shall consider the 

field at a point situated on the positive axis of y'. It is determined by 

all other components being 0. Since, neglecting terms of the second 
order, we may write 

d = d' - - [w h] 
instead of (98), we have 

d. = d;-^h;, 

from which it appears that the dielectric displacement takes the form 

d^ = acos{w(i;'- ^)+i)}. 
in which a is a constant. 



1) Note 25. 2) Note 26. 



6U I. GENEKAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

By substitution of the value of the local time and of the value (96) 
for y , this becomes 



d, = «cosj^(H--*^)(^-f)+p 



Thus we see that, at a fixed point of space, i. e. for a definite value 
of y, the frequency of the vibrations is given by 



•(1 +"'-)■ 



If the radiating particle has a positive velocity W , i. e. one that is 
directed towards the point considered, this frequency is higher than 
that of the particle itself, which, as is shown by (108), still has the 
value n. This is the well known change of frequency which, according 
to Doppler's principle, is caused by a motion of the source of light. 

46. Our second example relates to the reflexion of a beam of 
light by a perfectly reflecting mirror, for instance by one that con- 
sists of a pe3-fectly conducting substance. We shall suppose the 
incidence to be normal, and begin with the case of a mirror having 
no translation, so that we have to use the original equations. Let 
the beam of light be represented by (7) and let the surface of the 
mirror coincide with the plane YOZ. Then, the reflected beam, 
which we shall distinguish by the suffix (r), is given by 

tlj,(r) = — flJ cos n\t- +— ), ^,(r) = ^ cos nii +— j • 

Indeed, these values satisfy the condition that, at the mirror, there 
be no dielectric displacement along its surface. If we put a; = 0, 
we really find 

^y + f'.M = 0- 

The case of reflexion by a mirror moving with the velocity w^, 

in the direction of the axis of x, i. e. in the direction of its normal, 

can be treated by the same formulae, provided only we change x, t, d, h 

into X, t', d', h'.ij Therefore, if the incident beam is now represented by 

A^j = a cos nU'—^j, h; = a cos nit'—^, 

we shall have for the reflected light 

^'y{r) = — a COS w {t' + —^ , h'^^^, = a cos n it' + ^ . 

Let us now examine the values of d^,, h^, d^,;,.) and h^^^, in this case. 
The only component of w being w^, we flnd 

1) Note 27. 



REFLEXION BY A MOVING MIRROR. Gl 

so that the incident rays are given by 

h, = a(l + ^) eosn{t'-^-), 
and the reflected rays by 

^y(r) = - « (l -^ ^) COS W (t'+ j) , 

h.W = «(!-'?) cosw(r+^). 

In these formulae we shall now express t' and x in terms of t and x. 
The value of the local time is 

t'=t *"^' 



and 

re' = a; — yij. 
Hence 

'■-y = (i + ^)('-7). 

The formulae are simplified if we put 

a(l + ^) = a, nil+^^)^n. 

Continuing to neglect the square of — -, we infer from this 

so that the final formulae for the incident rays are 

d^ = a cosn(<— ^), h, = a cosn(i-yj, 
and those for the reflected light 

These equations show that both the frequency and the amplitude 
of the reflected beam are changed by the motion of the mirror. The 



62 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

frequency is now n(l ^\, smaller than n, if the mirror recedes 

from the source. These changes might have been predicted on the 
ground of Doppler's principle. As to the amplitude, it is changed 
in exactly the same ratio as the frequency, so that the reflected in- 
tensity is diminished by a motion in one direction and increased by 
a motion in the other direction. 

It is interesting to verify these results by considering the energy 
of the system. This may easily be done, if we iix our attention, 
not on the fluctuations of the electromagnetic energy, but on its 
mean value, so that, at every point of the beam, tv^ and w^ (§ 16) 
are considered as constants. Let the rays occupy a cylinder whose 
generating lines are parallel to OX and whose normal section is 2. 
Let P be a plane perpendicular to OX at some distance before the 
mirror and having a fixed position in the ether. If W^ is positive, 
so that the mirror i-ecedes, the space between the mirror and the 
plane P increases by W^2J per unit of time, so that the energy 
contained in that space increases by 

Again, if p is the pressure on the mirror, the work done by the 
field wiU be 

Consequently, if S is the current of energy towards the min-or per 
unit of area of the plane P, we must have 

Si:=(w,+ wJw^U + pv/^2J. (109) 

We can easily calculate the quantities occurring in this equation. 
In the incident beam there is a flow of energy (§ 17) 

towards the mirror, and in the reflected light a flow 

ia^(l-^-;fc = |a^(l-l|'.)c 

away from it, so that 

S = 2aV,. (110) 

As to w^, w^ and p, we may take for them the values that would 
hold if the mirror where at rest, because these quantities have to be 
multiplied by W^. Therefore, since the value of w^-^ tv^ consists of 
two equal parts, one belonging to the incident and the other to the 
reflected light*), 

^Ve + W'^ = al (111) 

1) Note 28. 



MOTION OF ELECTRICITY IN METALS. 63 

Finally, by what has been said in § 25, 

P = a2. (112) 

The values (110), (HI) and (112) really satisfy the condition (109). 

47. I shall close this chapter by a short account of the appli- 
cation of the theory of electrons to the motion of electricity in 
metallic bodies. In my introductory remarks, I have already alluded 
to the researches of Riecke, Drude and J. J. Thomson. I now 
wish especially to call your attention to the views that have been 
put forward by the second of these physicists. 

In his theory, every metal is supposed to contain a large number 
of free electrons, which are conceived to partake of the heat-motion 
of the ordinary atoms and molecules. Further, a well known theorem 
of the kinetic theory of matter, according to which, at a given tem- 
perature, the mean kinetic energy is the same for all kinds of pai-- 
ticles, leads to the assumption that the mean kinetic energy of an 
electron is equal to that of a molecule of a gas taken at the same 
temperature. Though the velocity required for this is very conside- 
rable, yet the electrons are not free to move away in a short time 
to a large distance from their original positions. They are prevented 
from doing so by their impacts against the atoms of the metal 
itself. 

For the sake of simplicity we shall assume only one kind of 
free electrons, the opposite kind being supposed to be fixed to the 
ponderable matter. Now, if the metal is not subjected to an electric 
force, the particles are moving indiscriminately towards all sides; 
there is no transfer of electricity in a definite direction. This changes 
however as soon as an electric force is applied. The velocities of the 
electrons towards one side are increased, those towards the other side 
diminished, so that an electric current is set up, the intensity of 
which can be calculated by theoretical considerations. The foi-mula 
to which one is led, of course contains the electric force, the number 
N of electrons per unit of volume, the charge e and the mass m of 
each of them. In the first place, the force acting on an electron is 
found if we multiply by e the electric force. Next dividing by m, we 
shall find the velocity given to the electron per unit of time. The 
velocity acquired by the electrons will further depend on the time 
during which they are exposed to the undisturbed action of the 
electric force, a time for which we may take the interval that elapses 
between two successive impacts against a metallic atom. During this 

interval, the length of which is given by — , if I is the path between 

the two encounters, and u the velocity of the electron, the electric 



64 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. 

force produces a certain velocity which we can take to be lost again 
at the next collision. 

These considerations will suffice for the explanation of the 
formula 



6 = 



imu ' 



which Drude has established for the electric conductivity of the 
metal, and in which we must understand by u the mean velocity of 
the electrons in their irregular heat -motion, and by I their mean 
length of free path. Now, as I said already, the mean kinetic energy 
of an electron, for which we may write ^mu\ is supposed to be 
equal to the mean kinetic energy of a gaseous molecule. The latter 
is proportional to the absolute temperature T, and may therefore be 
represented by 

uT, 

where a is a universal constant. If we use this notation, (113) takes 
the form 

a^^-^. (114) 



48. In order to show you all the beauty of Drude's theory, 
I must also say a few words about the conductivity for heat. This 
can be calculated in a manner much resembling that in which it is 
determined in the kinetic theory of gases. Indeed, a bar of metal 
whose ends are maintained at different temperatures, may be likened 
to a colum of a gas, placed, for example, in a vertical position, and 
having a higher temperature at its top than at its base. The process 
by which the gas conducts heat consists, as you know, in a kind of 
diffusion between the upper part of the column, in which we find 
larger, and the lower one in which there are smaller molecular velo- 
cities; the amount of this diffusion, and the intensity of the flow of 
heat that results from it, depend on the mean distance over which a 
molecule travels between two successive encounters. In Drude's 
theory of metals, the conduction of heat goes on in a way that is 
exactly similar. Only, the carriers by which the heat is transferred 
from the hotter towards the colder parts of the body, now are the 
fi-ee electrons, and the length of their free paths is limited, not, as 
in the case of a gas, by the mutual encounters, but by the impacts 
against the metallic atoms, which we may suppose to remain at rest 
nn account of their large masses. 

Working out these ideas, Drude finds for the coefficient of con- 
ductivity for heat 

k = ^aNlu. (115) 



CONDUCTIVITY FOR HEAT AND FOR ELECTRICITY. 65 

49. It is highly interesting to compare the two conductivities, 
that for heat and that for electricity. Dividing (115) by (114), 
we get 

"^-^[^P, (116) 

which shows that the ratio must be equal for all metals. As a rough 
approximation this is actually the case. 

We see therefore that Drude has been able to account for the 
important fact that, as a general rule, the metals which present the 
greatest conductivity for heat are also the best conductors of elec- 
tricity. 

Going somewhat deeper into details, I can point out to you two 
important verifications of the equation (116). 

In the first place, measurements by Jaeger and Diessel- 

horstM have shown that the ratio — between the two conductivities 

6 

varies approximately as the absolute temperature, the ratio between 

the values of — for 100" and 18" ranging, for the different metals, 

between 1,25 and 1,12, whereas the ratio between the absolute 
temperatures is 1,28. 

In the second place, the right-hand member of (116) can be 
calculated by means of data taken from other phenomena.^) In order 
to see this, we shall consider an amount of hydrogen, equal to an 
electrochemical equivalent of this substance, and we shall suppose 
this quantity to occupy, at the temperature T, a volume of one cubic 
centimetre. It will then exert a pressure that can easily be calculated, 
and which I shall denote by ]). 

We have already seen that the charge e which occurs in the 
formula (116), may be reckoned to be equal to the charge of an 
atom of hydrogen in an electrolytic solution. Therefore, the number 

of atoms in one electrochemical equivalent of hydrogen is — • The gas 

being diatomic, the number of molecules is -~, and the total kinetic 
energy of their progressive motion is 

aT 

2e 
per cubic centimetre. 

By the fundamental formula of the kinetic theory of gases the 



1) W. Jaeger und H. Diesselhorst, Warmeleitung, Elektrizitatsleitung, 
Warmekapazitat und Thermokraft einiger Metalle, Sitzungsber. Berlin 1899, p. 719. 

2) See M. Reinganum, Theoretische Bestimmung des Verhaltnisses von 
Warme- und Elektrizitatsleitung der Metalle aus der Drudeacheu Blektronen- 
theorie, Ann. Phys. 2 (1900), p. 398. 

Lorentz, Theory of electrons. 5 



66 I. GENERAL PRINCIPLES. THEORY OF PEEE ELECTRONS. 

pressure per unit area is numerically equal to two thirds of this, 
so that 

The equation (116) therefore takes the form 

or 



P 



= ]/,^|t. (117) 



This relation between the conductivities of a metal and other 
quantities derived from phenomena which, at first sight, have no con- 
nexion at all, neither with the conduction of heat, nor with that of 
electricity, has been verified in a very satisfactory way. 

The electrochemical equivalent of hydrogen being 

0,000104 

in our units, and the mass of a cubic centimetre of the gas at 0" 
and under a pressure of 76 cm of mercury being 0,0000896 gramni, 
one finds for the temperature of 18" (T = 273 + 1'^), 

la.Sxloi^ 



el/4 



7C 



On the other hand, expressing a in the ordinary electromagnetic units, 
Jaeger and Diesselhorst have found for silver at 18" 

A = 686 X 10^. 

G 

In our units this becomes 

Jc _ 686 X 108 

by which we find for the quantity on the right hand side of (117) 

12,9x 10' 

showing a very close agreement with the value we have just cal- 
culated for p. 

50. I must add, however, that the numerical agreement becomes 
somewhat less satisfactory, il^ instead of D rude 's formulae for the con- 
ductivities, one takes the equations to which I have been led by cal- 
culations that seem to me somewhat more rigorous than his. Taking 
into account that the electrons in a piece of metal have unequal velo- 



=v\ 



CONDUCTIVITY FOR HEAT AND FOR ELECTRICITY. 67 

cities, and assuming Maxwell's Jaw for the distribution of these 
among the particles, I find, instead of (114) and (115)^), 

and 

h = ^i\/^^ccmu. (120) 

In these equations, m is a velocity of such a magnitude that its square 
is equal to the mean square of the velocities which the electrons 
have in their heat-motion, and I represents a certain mean length of 
free path. 

The ratio of the two conductivities now becomes 

l = t(T)'^^ 

it is still proportional to the absolute temperature, but it is only 
two thirds of the value given by Drude. On account of this we 
must replace (117) by the equation 

whose right-hand side, in the example chosen in § 49, has the value 

15,8x10= 

This is rather different from (118). 

If we prefer the formulae (119) and (120) to (114) and (115), 
as I think we are entitled to do, the agreement found in the prece- 
ding paragraph must be considered as produced by a fortuitous coin- 
cidence. Nevertheless, even the agreement we have now found, cer- 
tainly warrants the conclusion that, in Drude's theory, a fair start 
has been made towards the understanding of the electric and thermal 
properties of metals. It is especially important to notice that our 
calculations rest on the assumption that the free electrons in a metal 
have charges equal to those of the ions of hydrogen. 

1) Note 29. 



CHAPTER 11. 

EMISSION AND ABSORPTION OF HEAT. 

51. The subject of this and my next lecture will be the radia- 
tion and absorption of heat, especially the radiation by what is called 
a perfectly black body, considered with regard to the way in which 
these phenomena depend on the temperature and the wave-length. 
I shall first recall to your minds the important theoretical laws which 
Kirchhoff, Boltzmann and Wien have found by an application 
of thermodynamic principles. After that, we shall have to examine 
how far the theory of electrons can give us a clue to the mechanism 
of the phenomena. 

We must begin by clearly defining what is meant by the absorb- 
ing power and the emissivity of a body. Let a and to' be two in- 
finitely small planes perpendicular to the line r joining their centres, 
and let Jf be a body of the temperature T, placed so that it can 
receive a beam of rays going through co' and a. We shall suppose 
this beam to consist of homogeneous rays whose wave-length is I, 
and to be plane-polarized, the electrical vibrations having a certain 
direction h, perpendicular to the line r. Part of the incident rays 
will be reflected at the front surface of the body, part of them will 
penetrate into its interior, and of these some will again leave the 
body, either directly or after one or more internal reflexions. How- 
ever this may be, the body M, if it be not perfectly transparent, 
will retain a certain amount of energy, an amount that is converted 
into heat, because we shall exclude from our considerations all other 
changes that might be produced. 

The coefficient of absorption A is defined as the fraction indi- 
cating what part of the incident energy is spent in heating the body 31. 

On the other hai:id, of the whole radiation emitted by 31, a 
certain portion will travel outwards through the two elements cj and co'. 
We shall decompose this radiation into rays of different wave-lengths, 
and we shall fix our attention on those whose wave-length lies be- 
tween two limits infinitely near each other, I and I -{- dX. We shall 
also decompose the electrical vibrations of these rays into a com- 



BMISSIVITY AND ABSORBING POWER. 69 

ponent along the line h of whicli I have just spoken, and a second 
component perpendicular both to it and to the direction of the beam 
itself It can easily be shown that the amount of energy emitted 
by the body per unit of time through the two elements of surface, 
so far as it belongs to rays of the wave-lengths that have been 
specified, and to vibrations of the direction /*, is proportional to 
CO, to', dl, and inversely proportional to the square of r. It can there- 
fore be represented by 

The coefficient E is called the emissivity of the body M. It is a 
quantity depending on the nature of M, its position with respect to 
the line r, the wave-length X, the temperature T and the direction h 
which we have chosen for the vibrations. 

Starting from the thermodynamic principle that in a system of 
bodies having all the same temperature, the equilibrium is not dis- 
turbed by their mutual radiation, and using a train of reasoning 
which I shall not repeat, Kirchhoff^) finds that the ratio 

E_ 
A 

between the emissivity and the absorbing power is independent, both 
of the direction we have chosen for h, and of the position and the 
peculiar properties of the body M. It will not be altered if we 
change th# position of 31, or replace it by an altogether different 
body of the same temperature. The ratio between the emissivity and 
the coefficient of absorption is a function of the temperature and the 
wave-length alone. 

52. I shall now point out to you two other meanings that may 
be attached to this function. In the first place, following the example 
of Kirchhoff, we can conceive a perfectly black body, or, as we shall 
simply say, a Hack body, i. e. one that has the power of retaining 
for itself the total radiating energy which falls upon it. Its coeffi- 
cient of absorption is therefore 1, and if we denote its emissivity 
by _Bj, the symbols A and E relating to any other body, we shall 
have 

1 = ^.- (122) 

We may notice in passing that Kirchhoff's law requires all black 
bodies, whatever be their nature, to have exactly the same emissivity. 



1) G. Kirchlioff, Uber das Verhaltnis zwischen dem Emissionsvermogen 
und dem Absorptionsvermogen der Korper fur Warme und Licht, Arm. Phys. 
Chem. 109 (1860), p. 275. 



70 n. EMISSION AND ABSORPTION OF HEAT. 

The equation (122) expresses one of the two meanings of j- 

to which I have alluded. The other will become apparent, if we fix 
our attention on the state existing in the ether in the neighborhood 
of radiating bodies. 

We shall consider a space void of all ponderable matter and 
surrounded on all sides by a perfectly black envelop, which is liept 
at a fixed temperature T. The ether within this space is traversed 
in all directions by rays of heat. Let a be an element of a plane 
situated at any point P of the space, and having any direction we 
like. We shall consider the quantity of energy by which this ele- 
ment is traversed per unit of time in the direction of its normal n, 
or rather in directions lying within an infinitely narrow cone, whose 
solid angle we shall denote by a, and whose axis coincides with 
the normal n, always confining ourselves to wave-lengths between X 
and X + dX, and to a particular direction h of the electrical vibrations. 
By this I mean that all vibrations of the rays within the cone are 
decomposed along lines h and h that are perpendicular as well to 
each other as to the axis of the cone, and that we shall only con- 
sider the components having the first named direction. 

Let P' be a point on the normal w, at a distance r from the 
point P, and let us place at P' an element of surface perpendicular 
to r, and whose magnitude is given by 

m'=r^s. (123) 

It is clear that, instead of speaking of the rays whose direction lies 
within the cone f, we may as well speak of those that are propagated 
through the elements a and ca'. 

The quantity we wish to determine is therefore the flow of 
energy through the two small planes, issuing from the part of 
the enclosing wall behind to. In virtue of the formula (121), it is 
given by 



for which, on account of (123), we may write 

E^coedX. (124) 

Having got thus far, we need no longer consider the element co'; 
we have only to think of the element co and the cone s. 

Now, what is most remarkable in our result, is the fact that it 
is wholly independent of the position of the point P, the direction 
of the element ro and the directions h and k, in which we have de- 
composed the vibrations. The radiation field within the ether is a 



ENERGY OF RADIATION PER UNIT OF VOLUME. 71 

truly isotropic one, i. e. the propagation takes place in exactly the 
same manner in all directions, and electrical vibrations of all different 
directions occur with the same intensity. 

We shall now calculate the amount of energy in this radiation 
field per unit of volume. In the case of a beam of rays of a definite 
direction the quantity of energy that is carried per unit of time 
through a plane a perpendicular to the rays, is equal to the amount 
existing at one and the same moment in a cylinder whose generating 
line\ as© parallel to the rays, and which has co for its base and a 
height equal to the velocity of light c; it is era times the energy 
existing per unit of volume. Hence, the energy per unit of volume, 
belonging to the rays to which the expression (124) relates, is found 
if we divide that expression by era; its value is 

c 

We must now keep in mind that we have all along considered only 
the rays whose direction lies within the cone s, and only those com- 
ponents of their vibrations which have the direction h. If we wish 
to include all rays, whatever be their direction and that of their 
vibrations, we must make two changes. In the first place we must 
multiply by 2, because the vibrations of the direction k have the 
same intensity as those we have till now considered, and in the se- 
cond place we must replace e by 4;r, because equal quantities of 
energy belong to rays whose directions lie within different cones of 
equal solid angles. The final result for the amount of energy present 
in unit volume of our radiation field, the „density" of the energy, 
so far as it is due to rays whose wave-lengths lie between the 
limits X and 2, ->r dl, is 

^E,dL 
c ° 

We shall write for this energy 

F{1, T')dX, 

so that, if we also take into account the relation (122), we have 

i^(Z,T) = ^i?, = i^f. (125) 

This equation, which expresses the relation between -.- and the 
density of energy, shows us the other meaning that may be given 

to -. • 

A 

53. One word more may be said about the state of radiation 
characterized by the function F(X, T). For the existence of this 



72 II. EMISSION AND ABSORPTION OF HEAT. 

state it is not at all necessary that the walls of the enclosure should 
be perfectly black. We may just as well suppose that they are per- 
fectly reflecting on the inside, and that the rays are produced by a 
body placed somewhere between them. Nor need this body be per- 
fectly black. Whatever be its nature, if it is maintained at the tem- 
perature T we have chosen once for all, it can always be in equili- 
brium with a state of radiation in which each element of volume 
contains the energy we have been considering. We may add that 
not only will it be in equilibrium with this state, but that it will 
actually produce it, provided only the body have some emitting 
power, however small it may be, for all wave-lengths occurring in 
the radiation of a black body of the same temperature. If this con- 
dition is fulfilled, the radiation in the ether will be independent of 
the nature of the matter in which it originates; it wiU be determined 
by the temperature alone. 

54. Kirchhoff has already laid stress on the importance of the 
function F{1, T), which must be independent of the peculiar pro- 
perties of any body, and indeed the problem of determining this 
function is of paramount interest in modem theoretical physics. 
Boltzmann^) and Wien^) have gone as far towards the solution as 
can be done by thermodynamic principles, combined with general 
results of electromagnetic theory, if one leaves aside aU speculations 
concerning the constitution of the radiating and absorbing matter. 

Boltzmann's law shows us in what way the total energy 
existing per unit of volume in the radiation field we have spoken of, 
I mean the energy for the rays of all wave-lengths taken together, 
depends on the temperature. It is proportional to the fourth power 
of the absolute temperature, a result that had already been established 
as an empirical rule by Stefan. 

In his demonstration, Boltzmann introduces the fact that there 
is a radiation pressure of the amount which we have formerly cal- 
culated. 

Let us consider a closed envelop, perfectly reflecting on the in- 
side, and containing a body-Jf to which heat may he given or from 
which heat may be taken, in one way or another. The remaining 
part of the space contains only ether, and the walls are supposed to 
be movable, so that the enclosed volume can be altered. 



1) L. Boltzmann, AMeitung des Stefan'sohen Gesetzes, betreiFend die 
Abhiingigkeit der Warmestrablung von der Temperatur aus der elektromagne- 
tisohen Lichttlieorie, Ann. Phya. Chem. 22 (1884), p. 291. 

2) W. Wien, Eine neue Beziehung der Strahlung schwarzer Korper zum 
zweiten Hauptsatz der Warmetheorie, Berlin. Sitzungsber. 1893, p. 55. 



BOLTZMANN'S LAW. 73 

The system we have obtained in this manner is similar in many 
respects to a gas contained in a vessel of variable capacity. It is 
the seat of a certain energy, and like a gas it exerts a pressure on 
the bounding walls; only, we have now to do, not with the collisions 
of moving molecules, but with the pressure of radiation. If the walls 
move outwards, the system does a certain amount of work on them. 
Hence, a supply of heat is required, if we wish to maintain a con- 
stant temperature, and the temperature is lowered by the expansion, 
if the process is adiabatic. You will easily see that the system may 
be made to undergo a cycle of operations, two of which are isother- 
mic and two adiabatic changes, and to which we may apply the well 
known law of Carnot. 

Instead of imagining a cycle of this kind, I shall use a small 
calculation that will lead us to the same result. In all cases in 
which the state of a system is determined by the temperature T and 
the volume v, and in which the only force exerted by the system is 
a normal pressure p uniformly distributed over the surface, there is 
a simple thermodynamic relation by which we can learn something 
about the internal energy s. If we choose v and T as independent 
variables, the equation has the form 

This may be applied to our envelop filled with rays, as well as to 
a gas; in a certain sense the case of the radiation is even the more 
simple of the two. The reason for this is, that the density of the 
energy depends solely on the temperature, so that, in an isothermic 
expansion, the new part that is added to the volume is immediately 
fiUed with an amount of energy proportional to its extent. The 
energy contained in the space that was already occupied by the radia- 
tion, remains unchanged, and the same may be said of the energy 
contained within the body M. In order to see this, we must keep 
in mind that, by what has been found in § 21, the pressure is equal 
to one third of the electromagnetic energy per unit of volume, so 
that the body remains exposed to the same pressure and, the tem- 
perature being likewise constant, will undergo no change at all. 

Let us denote by K the electromagnetic energy per unit of 
volume, which, as we must take together all wave-lengths, may be 
represented by 

CO 

K=fF{l,T)d}.. 



Then we shaU have 



74 n. EMISSION AND ABSORPTION OF HEAT, 

and 

because, if the Toliime is increased by dv, the energy augments by 
Kdv. Substituting in the formula (126), we find 











iK 


^dK 
^ dT' 










dK 
K 


A AT 
= 4 ^ , 


from 


which 


we 


deduce 


:by 


integration 
K=GT\ 



where is a constant. The total energy per unit of volume, or as, 
in virtue of (125), we may also say, the total emissivity of a black 
body must be proportional to the fourth power of the temperature. 

55. Passing on now to Wien's law, I shall first state the form 
in which it may be put if we avail ourselves of that of Boltzmann. 
Wien has not succeeded in determining the form of the function, 
which indeed cannot be done by thermodynamic reasoning and 
electromagnetic principles alone; he has however shown us how, as 
soon as the form of the function is known for one temperature, it 
may be found from this for any other temperature. 

This may be expressed as follows. If T and T' are two diffe- 
rent temperatures, X and 1' two wave-lengths, such that 

l:X'=T':T, (127) 

we shall have 

F{X, T) : F(r, T) = I's : X\ (128) 

If we put this in the form 

F{X',T')^^^-F(^X',T), 

we see that really F(X', T') can be determined for all values of X', 
if we know F{X, T) for all values of X. 

We can also infer from (127) and (12S) that if, while varying 
X and T, we keep the product XT constant, the function X^F(X, T) 
must also remain unchanged. Therefore, this last expression must 
be some function f{XT) of the product of wave-length and tempera- 
ture, so that our original function must be of the form 

F{X,T) = y{XT). (129) 



WIEN'S LAW. 75 

Tlie relation between the forms of the function F{X, T) for 
different temperatures comes out very beautifully. If, for a definite 
temperature T, we plot the values of F(X, T), taking I as abscissae 
and F as ordinates, we shall obtain a certain curve, which may be 
said to represent the distribution of energy in the spectrum of a 
black body of the temperature T. From this we can get the corre- 
sponding curve for the temperature T' by changing all abscissae in 
the ratio of T to T, and all ordinates in the ratio of T^ to T'l 

The form of the curve has been determined with considerable 
accuracy by the measurements of Lummer and Pringsheim.^) The 
accompanying figure wiE give an idea 
of it. It shows that, as could have 
been expected, the intensity is small 
for very short and very long waves, 
reaching a maximum for a definite 
wave-length which is represented by 
OA, and which I shall call 1^^. Now, 
if the curve undergoes the change of 
shape of which I have just spoken, 
this maximum will be shifted towards '^ '^si- 2 ^ 

the right if T' is lower than T, and 

towards the left in the opposite case, the value of l^ being in fact 
inversely proportional to the temperature. It is for this reason that 
Wien's law is often called the displacement- law (Verschiebungsgesetz). 

The diagram may also be used for showing that Boltzmann's 
law is included in the formulae (127) and (128). The value of K 
is given by the total area included between the curve and the axis 
of abscissae, and this area changes in the ratio of T^ to T'^, when 
the abscissae and ordinates are changed as has been stated. 

56. It would take too much of our time, if I were to give you 
a complete account of the theoretical deductions by which Wien found 
his law. Just as in Boltzmann's reasoning, we can distinguish 
two parts in it, one that is based on the equations of the electro- 
magnetic field, and a second that is purely thermodynamic. 

We have already seen that, for every temperature T, there is a 
perfectly definite state of radiation in the ether, which has the pro- 
perty that it can be in equilibrium with ponderable bodies of the 
temperature T. For the sake of brevity I shall call this the natural 



1) 0. Lummer u. E. Pringsheim, Die Strahlung eines schwarzen Korpers 
zwisolien 100 und 1300" 0, Ann. Phys. Chem. 63 (1897), p. 395; Die Verteilung 
der Energie im Spektrum des schwarzen Korpei-s, Verh. d. dentsohen phys. Ges. 
1 (1899), p. 28. 



76 U. EMISSION AND ABSORPTION OF HEAT. 

state of radiation for the temperature T. It is characterized by 
a definite amount K of energy per unit of Yolume, proportional 
to T^, and which may therefore be used, instead of T itself, for 
defining the state of the ether. If we speak of a natural state of 
radiation with the energy -density K, we shall know perfectly what 
we mean. 

In this natural state the total energy is distributed in a definite 
manner over the various wave-lengths, a distribution that is expressed 
by the function Fil, T). Now, we can of course imagine other 
states having the same density of energy K, but differing from the 
natural one by the way in which the energy is distributed over the 
wave-lengths; it might be, for example, that the energy of the long 
waves were somewhat smaller, and that of the short ones somewhat 
greater than it is in the natural state. 

Wien takes the case of a closed envelop perfectly reflecting on 
the inside, and containing only ether. He supposes this ether to be 
the seat of a natural state of radiation A with the energy-density K] 
this may have been produced by a body of the temperature T that 
has been temporarily lodged in the enclosure, and has been removed 
by some artifice. Of course, this operation would require a super- 
human experimental skill and especially great quickness, but we can 
suppose it to be succesfuUy performed. If then we leave the vessel 
to itself, the radiations that are imprisoned within it, will continue 
to exist for ever, the rays being over and over again reflected by the 
walls, without any change in their wave-lengths and their intensity. 

At this point, Wien introduces an imaginary experiment by 
which the state of things can be altered. It consists in giving to 
the walls a slow motion by which the interior volume is increased or 
diminished. We have already seen (§ 46) that, if a mirror struck 
normally by a beam of rays is made to recede, this will have a twofold 
influence on the reflected rays; their frequency is lowered, so that 
the wave-length becomes larger, and their amplitude is diminished. 
The same will be true, though in a less degree, if the incidence of 
the rays is not normal but oblique, and in this case also the effect 
can be easily calculated. 

In order to fix our ideas we shall suppose the walls of our 
vessel to expand. Then, every time a ray is reflected by them, it 
has its amplitude diminished and its wave-length increased, so that, 
after a certain time, we shall have got a new state of radiation B, 
differing from the original one by its energy per unit of volume and 
by the distribution of the energy over the wave-lengths. The den- 
sity of energy will have a certain value K', smaller than the original 
value K, and the distribution over the wave-lengths will have been 
somewhat altered in favour of the larger wave-lengths. 



WIBN'S LAW. 77 

Of course, K' can have different values, because the expansion by 
which the new state is produced may be a large or a small one. 
Since, however, the changes in the amplitudes and those in the wave- 
lengths are closely connected, it is clear that the distribution of the 
energy over the wave-lengths must be quite determinate if we know 
K', so that it was possible for Wien to calculate it. His result may 
be expressed as follows.') If 

(p{l)dX (130) 

is the part of the original energy per unit of volume that is due to 
the rays with wave-lengths between X and I + dX, the amount of en- 
ergy corresponding to the same interval in the new state B is given by 



n 



i) ■'p(Vi-^)^^- (131) 



57. I hope I have given you a sufficiently clear idea of one 
part of Wien's demonstration. As to the second part, the thermo- 
dynamic one, its object is to show that the new state B, in which 
there is a density of energy K', cannot be different from a natural 
state of radiation having the same K', that it must therefore itself 
be a natural state. If it were not, we could place our vessel con- 
taining the state B against a second vessel containing a natural state 
A' with the same value K', the two states being at first separated 
by the walls of the two vessels. Then we could make an opening 
in these walls, and close it immediately by means of a very thin plate 
of some transparent substance. Such a plate will transmit part of 
the rays by which it is struck, and, on account of the well-known 
phenomena of interference, the coefficient of transmission will not be 
the same for different kinds of rays. Let us suppose it to be some- 
what greater for the long waves than for the short ones, and let us 
also assume that the state B contains more of the long waves than 
the state A', and less of the short waves. Then, it is easily seen that, 
in the first instants after communication has been established between 
the two vessels, more energy wiU pass from B towards A' than in 
the inverse direction, so that the energy of the two states will not 
remain equal. This can be shown to be in contradiction with the 
second law of thermodynamics. 

Our conclusion must therefore be that, by means of the ex- 
pression (131), we can calculate the distribution of energy in a natural 
state characterized by K', as soon as we know the distribiition, re- 
presented by (130), for a natural state characterized by K. Now, 
both states being natural ones, we shall have, if we write T and 1" 



1) Note 30. 



78 11. EMISSION AND ABSORPTION OF HEAT. 

for the temperatures to which they correspond, 

K : K' = T^ : T'\ 
Therefore, (131) becomes 

by which we are led to Wien's law in the form in which I have 
stated it. 

58. Though Boltzmann and Wien have gone far towards 
determining the function F{X, T), the precise form of the curve in 
Fig. 2 remains to be found, and since the means of thermodynamics 
are exhausted, we can only hope to attain this object, if we succeed 
in forming some adequate mental picture of the processes which mani- 
fest themselves in the phenomena of radiation and absorption. 

The importance of the problem will be understood, if one takes 
into account that the curve in Fig. 2 requires for its determination 
at least two constants. Calling /l„, the abscissa OA for which the 
ordinate is a maximum, we have by Wien's law 

X ="- 

and if, as before, the total area included between the curve and the 
axis of abscissae is denoted by K, we shall have 

Of the two constants a and h, the first determines, for a given tem- 
perature T, the position of the point A, and the second relates to 
the values of the ordinates, because the larger these are, the greater 
will be the area K. Now, if the state of radiation is produced by a 
ponderable body, the values of the two constants must be determined 
by something in the constitution of this body, and these values can 
only have the universal meaning of which we have spoken, if all 
ponderable bodies have something in common. If we wish comple- 
tely to account for the form and dimensions of the curve, we shall 
have to discover these common features in the constitution of all 
ponderable matter. 

59. I shall speak of three theories by which the problem has 
been at least partially solved, beginning with the one that goes farthest 
of all. This has been developed by Planck^), and leads to a definite 
formula for the function f(lT) in (129), viz. to 

1) M. Planck, Uber irreversible Strahlungsvorgange, Ann. Phys. 1 (1900), 
p. 69; tjber daa G-esetz der Knergieverteilung im Normalspektrum, Ann. Phys. 4: 
(1901), p. 553; Tiber die Elementarquanta der Materie und der Elektrizitiit, 
ibid., p. 564; see also his book: Vorlesungen uber die Theorie der Warme- 
strahlung, Leipzig, 1906. 



THEORY OF PLANCK. 79 

F{X,T) = '-^.^-^, (132) 

kXT 
S —1 

in which e is the basis of natural logarithms, whereas h and h are 
two universal physical constants. 

Planck's theory is based on the assumption that every ponde- 
rable body contains an immense number of electromagnetic vibrators, 
or „resonators" as he calls them, each of which has its own period. 
If a body is enclosed within the perfectly reflecting walls we have 
so often mentioned, there will be a state of equilibrium, on the one 
hand between the resonators and the radiation in the ether, and on 
the other hand between the resonators and the ordinary heat motion 
of the molecules and atoms constituting the ponderable matter. The 
first of these equilibria can be examined by means of the electro- 
magnetic equations, and, in order to understand the second, one 
could try to trace the interchange of energy between the resonators 
and the ordinary particles. Planck, however, has not followed this 
course, which would lead us into very serious difficulties, but has 
found his formula by reasonings of a different kind. 

In one of his papers he deduces it by examining what partition 
of the energy between the two sets of particles, the molecules and 
the resonators, is to be considered as the most probable one. Of 
course this is an expression, the precise meaning of which has to be 
fixed before we can make it the basis of the theory. I must abstain 
from explaining the sense in which it is understood by Planck. 
There is one point, however, in his theory to which I must refer for 
a moment. He is obliged to assume that the resonators can gain 
or lose energy, not quite gradually by infinitely small amounts, but 
only by certain portions of a definite finite magnitude. These por- 
tions are taken to be difi^erent for resonators of different frequencies. 
The portions of energy which we have to imagine when we speak 
of a resonator of the frequency n, have an amount that is given 

by the expression 

Jin 

It is in this way that the constant li is introduced into the equations. 

As to the constant /,, it has a very simple physical meaning. 
According to the kinetic theory of gases, the mean kinetic energy of 
the progressive motion of a molecule is equal for all gases, when 
compared at the same temperature. This mean energy is proportional 
to T, and if we represent it by f/cT, the quantity k will be the 
constant appearing in the formula (132). 

Planck's law shows a most remarkable agreement with the 
experimental results of Lummer and Pringsheim, and it is of 



80 II. EMISSION AND ABSORPTION OF HEAT. 

high value because it enables us to deduce from measurements on 
radiation the mean kinetic energy of a molecule, which, in its turn, 
leads us to the masses of the atoms in absolute measure. As the 
numbers obtained in this way^) are of the same order of magnitude 
as those that have been found by other means, there is undoubtedly 
much truth in the theory. Yet, we cannot say that the mechanism 
of the phenomena has been unveiled by it, and it must be admitted 
that it is difficult to see a reason for this partition of energy by 
finite portions, which are not even equal to each other, but vary 
from one resonator to the other. 

60. I shall dwell somewhat longer on the second theory^), be- 
cause it is an application of the theory of electrons, and therefore 
properly belongs to my subject. In a certain sense, it may, I think, be 
considered as rather satisfactory, but it has the great defect of being 
confined to long waves. I may be permitted perhaps, by way of in- 
troduction, to teU. you by what considerations I have been led to 
this theory. It is well known that, in general, the optical properties 
of ponderable bodies cannot be deduced quantitatively with any 
degree of accuracy from the electrical properties. For example, though 
Maxwell's theoretical inference, published long ago in his treatise, 
that good conductors for electricity must be but little transparent 
for light, is corroborated by the fact that metals are very opaque, 
yet, if we compare the optical constants of a metal, one of which is 
its coefficient of absorption, with the formulae of the electromagnetic 
theory of light, taking for the conductivity the ordinary value that 
is found by measurements on electric currents, there is a very wide 
disagreement. This shows, and so does the discrepancy between tbe 
refractive indices of dielectrics and the square root of their dielectric 
constants, that, in the case of the very rapid vibrations of light, cir- 
cumstances come into play with which we are not concerned in our 
experiments on steady or slowly alternating electric currents. 

If this idea be right, we may hope to find a better agreement, 
if we examine the „optical" properties as we may continue to call 
them, not for rays of light, but for infra-red rays of the largest 
wave-lengths that are known to exist. 

Now, in the case of the metals, this expectation has been verified 
in a splendid way by the measurements of the absorption that were 
made some years ago by Hagen and Rubens.') These pbysicists have 

1) Note 31. 

2) Lorentz, On the emission and absorption by metals of rays of heat 
of great wave-lengths, Amsterdam Proc, 1902 — 03, p. COG. 

3) E. Hagen u. H. Rubens, Uber Beziehungen des Reflexions- und 
Emissionsvermogens der Metalle zu ihrem elektrischen Leitvermogen Ann Phvs 
11 (1903), p. 873. ■ J ■ 



ABSORPTION BY A THIN METALLIC PLATE. 81 

shown that rays whose wave-length is between 8 and 25 microns, are 
absorbed to a degree that may be calculated with considerable accuracy 
from the known conductivity.^) We can conclude from this that, in 
order to obtain a theory of absorption in the case of these long waves, 
we only have to understand the nature of a common current of con- 
duction. Moreover, if in this line of thought, we can form for our- 
selves a picture of the absorption, it must also be possible to get an 
insight into the way in which rays are emitted by a metal. Indeed, 
the universal validity of Kirchhoff's law clearly proves that the 
causes which produce the absorption by a body, and those which call 
forth its radiation, must be very closely related. Therefore, as soon 
as we have an adequate idea about a common current of conduction, 
we may hope to be able to explain the absorption and the emissivity 
of a metal, and to calculate the ratio between the two, i. e. our uni- 
versal function F{1, T). However, we can only hope to succeed in 
this, if we confine ourselves to long waves. 

Now, as we have already seen, a very satisfactory conception of 
the nature of a current of conduction has been worked out by 
Drude. We must therefore try to obtain a theory of the radiation 
and emission of metals that is based on his general principles, and 
in which we simply assume that the metal contains a large number 
of free electrons, moving with such speeds that their mean kinetic 
energy is equal to uT. 

61. In doing so, we shall simplify as much as possible the cir- 
cumstances of the case. We shall consider a metal plate, whose thickness 
z/ is so small that the absorption may be considered as proportional to 
it, and that, in examining the emission, we need not consider the ab- 
sorption which the rays emitted by tbe back half of the plate undergo, 
while traversing the layers lying in front of it. We shall also confine 
ourselves to rays whose direction is perpendicular to the plate or makes 
an infinitely small angle with the normal. These assumptions wiU 
greatly facilitate our calculations without detracting from the generality 
of the final result. If we trust to Kirchhoff's law, the value which 
we shall find for the ratio between the emissivity and the coefficient 
of absorption may be expected to hold for all bodies and for all 
directions of the rays. 

The calculation of the absorption is very easy. By the ordinary 
formulae of the electromagnetic field we find for the coefficient of 
absorption^) 

A = ~ J, 



1) Note 32. 2) Note 33. 

Lorentz, Theory of electrons. 



82 11- EMISSION AND ABSORPTION OF HEAT. 

and here we have only to substitute the value of 0, given by Drude's 
theory. Using the formula (119), we find 

r S7t acT ^ 

62. The question now arises, in what manner a piece of metal 
in which free electrons are moving in all directions can be the source 
of a radiation. The answer is contained in what we have seen in a 
former lecture. We know that an electron can be the centre of an 
emission of energy only when its velocity changes. The cause of 
the emission must therefore be looked for in the impacts against the 
metallic atoms, by which the electron is made to rebound in a new 
direction, so that the radiation of heat, in the case we are now con- 
sidering, very much resembles the production of Rontgen rays, as 
it is explained in Wiechert's and J. J. Thomson's theory. 

The mathematical operations required for the determination of 
the effects of the impacts are rather complicated, the more so because 
we must decompose the total radiation into the parts corresponding 
to different wave-lengths. I shall therefore give only a general outline 
of the calculations. 

I must mention in advance that the decomposition of which 
I have spoken just now will be performed by means of Fourier's 
theorem, and that the duration of an impact will be taken to be ex- 
tremely small in comparison with the time of vibration of the rays 
considered. We shall even make the same assumption with regard 
to the time between two successive impacts of an electron. This is 
justified by the experiments of Hagen and Rubens. It is easily 
seen that the conductivity of a metal can be given by the formula 
(119), only if the electric force acts on the body either continually 
or at least for a time during which a large number of encounters of 
an electron take place. Therefore, the result found by Hagen and 
Rubens, viz. that the absorption corresponds to the coefficient of 
conductivity, proves that the time during which the electric force 
acts in one and the same direction, i. e. half a period, contains very 
many times the interval between two successive encounters. 

63. In § 51 we have considered the radiation from the body M 
through two infinitely small planes cd and a'. We shall now suppose 
the first of these to be situated in the front surface of the thin 
metallic plate, and we shall fix our attention on the radiation issuing 
from the corresponding part co J of the plate, and directed towards 
the element ra', parallel to m^ and situated at a point P of the line 
drawn normally to the plate from the centre of the element a. 



EMISSION BY A THIN PLATE. 83 

We shall begin by taking into account only the component of the 
electric vibrations in a certain direction h perpendicular to OP. 

Let us choose the point as origin of coordinates, drawing the 
axis of z along OP, that of a; in the direction h, and denoting the 
distance OP by r. According to what has been found in § 39, 
a single electron, moving with the velocity V in the part of the plate 
considered, will produce at P a dielectric displacement whose first 
component is given by 

e dv^ 
inc^r dt ' 

if we take the value of the differential coefficient for the proper 
instant. 

On account of our assumption as to the thickness of the plate, 
this instant may be represented for all the electrons in the portion caz/ 

by ^ , if i is the time for which we wish to determine the state 

of things at the point P. We may therefore write for the first 
component of the dielectric displacement at P 

d^ = -^V(^e%V .. (134) 

c 

The flow of energy through m per unit of time will be 

cd>'. 

Since the motion of the electrons between the metallic atoms is 
highly irregular, we shall have, at rapidly succeeding instants, a large 
number of impacts in which the changes of the velocity are widely 
different. The state at P, which is due to aU these impacts, will 
show the same irregularity. Nevertheless, we must try to deduce 
from the formulae relating to it, results concerning those quantities 
that can make themselves felt in actual experiments. 

Results of this kind are obtained by considering the mean values 
of the variable quantities calculated for a sufficiently long lapse of 
time. We shall suppose this time to extend from ^ = to t = d: 
If the mean value of d| is denoted by d;, we shall have for the flow 
of energy through a that is accessible to our means of observation 

,9- 

c^im'^ca'-^fAldt. (135) 



64. The introduction of this long time &■ is also very useful 
for the application of Fourier's theorem. Whatever be the way in 
which d^ changes from one instant to the next, we can always expand 
it in a series by the formula 



84 n. EMISSION AND ABSORPTION OF HEAT. 

d. = 2«.sinlj, (136) 

where s is a positive whole number, each coefficient a^ being deter- 
mined by 

«^ = iA¥M^- (137) 



It appears from (136) that the frequency of one of the terms is 

Sit 



so that the corresponding wave-length is given by 

2«c 2c«' 



(138) 



The interval -9- being very large, the values of I belonging to small 
values of s will be so too; we shall not, however, have to speak of 
these very long waves, because they may be expected to represent 
no appreciable part of the total radiation. The rays with which we 
are concerned, will have wave-lengths below a certain upper limit l^,; 
therefore, provided the time -9' (which we are free to choose as long 
as we like) be long enough, they will correspond to very high values 
of the number s. Now, if 1^ and l^_^^ are two successive wave- 
lengths, we shall have 



!+l S + 1' 

which is a very small number. The wave-lengths corresponding to 
the successive terms in our series are thus seen to diminish by ex- 
ceedingly small steps. This means that, if we were to decompose 
the radiation represented by (136) into a spectrum, we should find a 
very large number of lines lying closely together. Their mutual 
distances may be indefinitely diminished by increasing the length of 
the time %■ and the values of s corresponding to the part of the 
spectrum we wish to consider. This is the way in which we can 
deduce from our formulae the existence of a continuous spectrum 
and the laws relating to it. 

Let I and I + dl be two wave-lengths, which, from a physical 
point of view, may be said to lie infinitely near each other. If %■ 
is duly lengthened, the part of the spectrum corresponding to dl 
contains a large number of spectral lines, for which we find 

-^ dl. 



EMISSION BY A THIN PLATE. 85 

This is clear, if, after haying written (138) in the form 

we observe that the number of lines is the same as the number of 
integers lying between the limits 

and 



for which we may take the difference 

2cfl- 

because, in virtue of our supposition, this difference is much larger 
than 1. 

We have now to substitute the value (136) in the equation (135). 
It is easily seen that the product of two terms of the series for d 
will give 0, if integrated with respect to time between the limits 
and &. Moreover 



and (135) becomes 



cd>'=ic«'2«f- (139) 



= 1 



This is the total flow of energy through a'. In order to find the 
part of it, corresponding to wave-lengths between I and A + dX, we 

have only to observe that the -jr^^^ spectral lines lying within that 

interval, may be considered to have equal intensities.^) 

In other terms, the value of a^ may be regarded as equal for 
each of them, so that they contribute to the sum in (139) an amount 

Consequently, the part of the flow of energy, belonging to the inter- 
val of Avave-lengths dl, is given by 

"^'^-a^dl, (140) 

and our problem wiE be solved, if we succeed in calculating a^. 

65. The following mathematical developments are somewhat 
more rigorous than those which I gave in my paper on the subject. 

1) Note M. 



86 n. EMISSION AND ABSORPTION OF HEAT. 

In fact, I sliall now introduce Maxwell's law for the distribution 
of the velocities among the electrons, and take into account that the 
free paths are not all of the same length. At the same time I shall 
introduce a simplification for which I am indebted to Langevin'), and 
by which it will be possible to give in a small space the essential 
part of the calculation. 

By (134) and (137) we see that 



■inO-c'^r ^ 



(«/--:' -"2'' *i. (i«) 



where the square brackets serve to indicate the value of V^ at the 



T 

time t 

c 



The meaning of this equation is, that we must first, for one 
definite electron,, calculate the integral, taking into account all the 
values of the acceleration occurring during the interval of time 

between and ■9- This havinsr been done, we have to take 

the sum of the values that are found in this way for all the free 
electrons contained in the part C3z/ of the plate. 

Integrating by parts we find, because sin — ^ vanishes at the 
limits. 



"» 2«'^c2»- 



2'l/Wcos^-^^), (142) 



for which, imderstanding by V^ the value at the time i, we may also 
write 






/'.°«f('+o-) 



dt 



By this artifice of partial integration, the problem is reduced to a 
much simpler one. If we had directly to calculate the integral 
in (141), we should have to attend to the intervals of time durino- 
which an electron is subjected to the force which makes it rebound 
from an atom against which it strikes; indeed, it is only during these 
intervals that there is an acceleration. On the other hand, the inte- 
gral in (142) is made up of parts, due, not only to the times of 

1) See his translation of my paper in H. Abraham et P. Langevin, Les 
quantit^s eWmentaires d'^lectricite, ions, electrons, oorpuscules, Paris (1905) 1 
p. 507. ' 



EMISSION BY A THIN PLATE. ,S7 

impact, but also to all intervening intervals. If we suppose the du- 
ration of an encounter to be very much smaller than the lapse of 
time between two successive collisions of an electron, we may even 
confine ourselves to the part that corresponds to the free paths be- 
tween these collisions. 

While an electron travels over one of these free paths, its velo- 
city V^ is constant. We may also neglect the change in the factor 

cos^^^ +~j, because the time between two encounters is supposed 

to be very much smaller than the time of vibration corresponding 
to s. The part of a^ which corresponds to one electron, and to 
the time during which it describes one of its free paths, is therefore 
given by 

if we understand by r the time during which the path is travelled 
over. In the last factor we may take for t the value corresponding 
to the middle of the time r. 

We shall now fix our attention on all the paths described by 
all the electrons during the time Q: If we use the symbol S for 
denoting a sum relating to all these free paths, we shall have 

rV,cos^(^-f-f)). (143) 



se 



aS'^cV 



66. We have to determine the square of the sum S. This may 
be done rather easily, because the products of two terms 



S7C 

rv^ cos -r- 



('H). 



whether they correspond to two different free paths of one and the 
same electron, or to two paths described by different electrons will 
give 0, if all taken together. Indeed, the velocities of two electrons 
are wholly independent of each other^ and the same may be said of 
the velocities of one definite electron at two instants between which it 
has undergone one or more impacts.') Therefore, positive and negative 
values of V^ being distributed quite indiscriminately between the terms 
of (143), positive and negative signs will be equally probable for the 
products of two terms. 

It is seen in this way that we have only to calculate the sum 
of the squares of the several terms, so that we find 

«»=I^«h^vIcos^f(.-Ff)). 
1) Note 35. 



88 11- EMISSION AND ABSORPTION OF HEAT. 

Now, since the irregular motion of the electrons takes place with the 
same intensity in all directions, we may replace v| by ^V^ Therefore, 
writing I for the length t | V | of the free path, we find 



3 s'e 



;^.s(Pcos^f(^+-^)) 



In the immense number of terms included in the sum, the length I 
is very different, and in order to effect the summation we may 
begin by considering only those terms for which it has a certain 
particular value. In these terms, which are still very numerous, the 

angle -^ (t -] i has values that are distributed at random over an 

interval ranging from to sn. The square of the cosine may there- 
fore be replaced by its mean value \, so that 

67. The metallic atoms being considered as immovable, the 
velocity of an electron is not altered by a collision. We can there- 
fore fix our attention on a certain group of electrons which move 
along their zigzag-lines with a definite velocity u. During the time %•, 
one of these particles describes a large number of free paths, this 
number being given by 

u& 

if ?„j is the mean length of the paths. It can be shown ^) that the 
number of paths whose length lies between I and I + dl, is 



so that 



-^ - s 'ii, dl. 



— ^2 — ' ^ ^"^ di 



is the part of the sum S(Z^) contributed by these paths. Integrating 
with respect to I from to oo, we find 

2^mL (145) 

for the value of S(P) in so far as it is due to one electron. 

The total number of electrons in the part of the metallic plate 
under consideration is Nm/I and, by Maxwell's law, among these 

43tiV(DZ?]/|^£-»"V(?M (146) 



1) Note 36. 



RATIO BETWEEN BMISSIVITY AND ABSORPTION. 89 

have velocities between u and u + du, the constant q being related 
to the velocity m,„ whose square is equal to the mean value of u^, 
by the formula 

a 
3 = 



2m|. 



In order to find the total value of S(V) we must multiply (145) by 
(146), and integrate the product between the limits m = and m = oo. 
Supposing ;,„ to be the same for all values of m^), we find 

Finally, the equation (144) becomes 



«5 = l/al- 






and the expression (140) for the radiation through the element oj' 
takes the form 

or, in virtue of (138), if, instead of ?„,, m„,, we simply write Z, u, 

'V 2 eHNu 



rH' 



ma JdX. 



This is the energy radiated per unit of time, in so far as it belongs 
to wave-lengths between X and X + dX, and to the components of 
the vibrations in one direction }i. Thus, the quantity we have cal- 
culated is exactly what was represented by (121), and on comparing 
the two expressions we find 

for the emissivity of the plate. 

68. We have now to combine this with the value (133), which 
we have found for the coefficient of absorption. If Kirchhoff's 

law is to hold, the ratio —^ must be independent of those quantities 

by which one metallic plate differs from the other. This is really 
seen to be the case, since the number iV of electrons per unit of 
volume, the mean length I of theii- free paths and the thickness A 

of the plate aU disappear from the ratio. We really get for -^ and 

for F{X, T) values that are independent of the peculiar properties of 



1) Note 37. 



kXT 
E = 


1 + 


ch 


F{1, T) 


8 


TtlcT 



90 II. EMISSION AND ABSORPTION OF HEAT. 

any ponderable body. I must repeat however that all our considerations 
only hold for large wave-lengths. 

Using the formulae (125), (133) and (147), we find 

i^(A,T) = ^^. (148) 

It is very remarkable that this result is of the form (129) and that 
it agrees exactly with that of Planck. This may be seen, if in (132) 
we suppose the product AT to have a very large value, so that the 
exponent is very small. Then, we may put 



and (132) becomes 



This is equal to (14H), because our coefficient a corresponds to |fc 
in Planck's notation. As has been stated, the mean kinetic energy 
of a molecule of a gas is -^cT, and we have represented it by ccT}) 

69. A widely different theory of the radiation of a black body 
has been developed by Jeans. ^) It is based on the theorem of the 
so-called equipartition of energy, which plays an important part in 
the kinetic theory of gases and in molecular theories in general. 
In its most simple form it was discovered by Maxwell in 1860; 
afterwards it was largely extended by Boltzmann, and Jeans has 
given an ample discussion of it in his book on the kinetic theory 
of gases. 

Maxwell was led to the theorem by his theoretical investigations 
concerning the motion of systems consisting of a large number of 
molecules. If, from a mass of gas, we could select single molecules, 
we should find them to move with very different velocities, and to 
have very different kinetic energies. The mean kinetic energy of the 
progi-essive motion, taken for a sufficiently large number of molecules 
will however be the same in adjacent parts of the gas, if the tem- 
perature is the same everywhere, so that these parts can be said to 
be in equilibrium. This wiU even be true if the gas is subjected to 
external forces, such as the force of gravity, which make the density 
change from point to point. Also, if we have a mixture of two 
gases, the mean kinetic energy of a molecule can be shown to be 
equal for the two constituents, and we can safely assume that for 



1) Note 38. 

2) J. H. Jeans, On the partition of energy between matter and aether, 
Phil. Mag. (6) 10 (1905), p. 91. 



EQUIPAETITION OF ENERGY. 91 

two gases that are not mixed, but kept apart, this equality of the 
mean kinetic energy ol a molecule is the condition for the existence 
of equilibrium of temperature. We can express this by saying that 
the kinetic energy of a gas, in so far as it is due to the progressive 
motion of the molecules, can be calculated by attributing to each 
molecule an amount of energy having the same definite value, 
whatever be the nature of the gas. 

This amount of energy is proportional to the absolute tempera- 
ture T, and may therefore be represented, as I have done already 
several times, by aT, a being a universal constant. 

We can express the result in a somewhat different way. If the 
molecules of the gas are supposed to be perfectly elastic and rigid 
smooth spheres, the only motion with which we are concerned in 
these questions is their translation; the position of the particles can 
therefore be determined by the coordinates x, y, 2 of their centres. 
If T is the number of molecules, the configuration of the whole 
system requires for its determination 3^^' coordinates, or, as is often 
said, the system has 3i\" degrees of freedom. To each degree of 
freedom, or to each coordinate x, tj or z, corresponds a certain velo- 
city X, y or 2, and also a certain kinetic energy ^-mx-', \mi/, ^}mk^. 
The total energy of the gas can be calculated by taking ^aT for 
the kinetic energy corresponding to each degree of freedom. The 
factor -^ is here introduced because the total kinetic energy of a 
molecule, whose mean value is aT, is the sum of the quantities 
^mx^, i-mi/^, l-mz^, corresponding to its three degrees of freedom. 

70. These remarks will suffice for the understanding of what 
is meant by the equipartition of energy in less simple cases. The 
configuration of a body of any kind, i. e. the position of the ultimate 
particles of which it is conceived to be made up, can always be de- 
termined, whatever be the connexions between these particles, by a 
certain number of coordinates p in the general sense in which the 
term has been used by Lagrange, and these coordinates can often 
be chosen in such a manner that the kinetic energy is equal to a 
sum of terms, each of which is proportional to the square of one of 
the velocities p, so that it may be said to consist of a number of 
parts corresponding to the different degrees of freedom of the system. 
The theorem of equipartition teUs us that, if the temperature is T, 
the kinetic energy of a system having a very large number of degrees 
of freedom, as all bodies actually have, can be found by attributing 
to each degree of freedom a kinetic energy equal to ^ccT. 

It should be noticed that it is only the kinetic energy that can 
be calculated in this way. If we wish to determine the whole energy, 
we must add the potential part of it. Now, there is one case, and 



92 n. EMISSION AND ABSORPTION OF HEAT. 

it is the very one that is most relevant to our purpose, in vs^hich 
the value of the potential energy is likevrise determined by a very 
simple rule. 

Let us consider a system capable of small vibrations about a 
position of stable equilibrium, and let the coordinates Pi, p^, ■ ■■> Pn 
be in this position, so that they measure the displacement of the 
system from it. These coordinates can be chosen in such a way that 
not only, as we have already required, the kinetic energy is the sum 
of a number of terms each containing the square of a velocity p, but 
that, besides this, the potential energy is expressed as a similar sum 
of terms of the form ap'^, where « is a constant. 

The most general motion of the system is made up of what we 
may call fundamental or principal modes of vibration. These are 
characterized by the peculiarity that in the first mode only the co- 
ordinate p^ is variable, in the second only p^, and so on, the variable 
coordinate being in every case a simple harmonic function of the 
time t, with a frequency that is in general different for the different 
modes. It is a fundamental property of these principal vibrations, 
that, in each of them, the mean value of the potential energy for a 
full period, or for a lapse of time that is very long in comparison 
with the period, is equal to the mean value of the kinetic energy. 
Moreover, if the system vibrates in several fundamental modes at the 
same time, the total energy is found by adding together the values 
which the energy would have in each of these modes separately.^) 

71. We shall now suppose a system of this kind, having a very 
large number of degrees of freedom, to be connected with an ordinary 
system of molecules, with a gas for example, so that it can be put 
in motion by the forces which it experiences from the molecules, and 
can in its turn give off to these a part of its vibratory energy. Then, 
there can be a state of equilibrium between the heat motion of the 
molecules and the vibratory motion of the system. We may even 
speak of the vibrations of the system as of its heat motion, and say 
that the system has a definite temperature, the same as that of the 
system of molecules with which it is in equilibrium. 

The theorem of the equipartition of energy requires that, whatever 
be the exact way in which the vibrating system loses or gains energy, 
it shall have for each of its coordinates a kinetic energy given by 
\aT. The sum of the potential and kinetic energies must be -|o;T 
for each of its fundamental modes of vibration, and the problem of 
determining the total energy is, after all, a very simple matter. We 
need not even specify the coordinates by which the configuration of 

1) Note 39. 



JEANS'S THEORY OF RADIATION. 93 

the system can be determined. All we want to know is the number 
of the fundamental modes of vibration; multiplying by this the quan- 
tity faT, we shall have the energy of the system corresponding to 
the temperature T. 

72. It was a most happy thought of Jeans to apply this 
method to the problem of radiation. It enables us to calculate 
the energy of radiation in the ether for a certain temperature T 
without having to trouble ourselves about the mechanism of emission 
and absorption, without even considering a ponderable body. The 
only question is, what is the number of degrees of freedom for a 
certain volume of ether. For the sake of convenience we shall enclose 
this volume by totally reflecting walls, and to begin with, we shall 
imagine two such walls, unlimited parallel planes at a distance q from 
each other. The ether between them can be the seat of standing 
waves, which we can compare to those existing in an organ-pipe, and 
which may be conceived to arise from the superposition of systems 
of progressive waves. 

The condition at a perfectly reflecting surface is that Poyn ting's 
flow of energy be tangential to it. It will be so if, for example, the 
surface is a perfect conductor, the tangential components of the 
electric force being in this case. Let us suppose the two boundary 
planes to be of this kind. If they are perpendicular to the axis of x, 
their equations being a; = 0, and x = q, the condition for the electric 
force can be fulfilled by the superposition of two sets of progressive 
waves, such as are represented by the equations (7) and by those 
given in § 46. The total dielectric displacement 



a cos n(t \ — a cos nit -\ j 



= 2a sLq nt sin 



nx 
c 



11 q 



will be for a; = 0, and also for x = q, if is a multiple of n, 



c 



or, what amounts to the same thiQg, if the distance g is a multiple 
of half the wave-length. The possible modes of motion will therefore 
have wave-lengths equal to 2q, q, -|g, etc. 

73. We shall next examine the vibrations that can take place 
in the ether contained within a box, whose walls are perfectly reflecting 
on the inside, and which has the form of a rectangular parallelepiped. 
Let the axes of coordinates be parallel to the edges, and let the lengths 
of these be q^, q^, q^. 

We can imagine eight lines such that their direction constants 
have equal absolute values, but aU possible algebraic signs; indeed. 



94 II. EMISSION AND ABSORPTION OP HEAT. 

denoting by fi^, fi^, (ig the absolute values of the constants, we shall 
have the eight combinations 



(-f^lj — ^2,-^3) • 



(149) 



If a beam of parallel rays within the rectangular box has one of these 
lines for its direction of propagation, the reflexion at the walls wiU 
produce bundles parallel to the other seven lines, and if the values 
of (i^, fi.j, jiig and the wave-length X are properly chosen, the boundary 
conditions at the walls can be satisfied by the superposition of eight 
systems of progressive waves travelling in the eight directions. In 
order to express the condition to which fij, /i^, (i^, I must be sub- 
jected, we shall imagine three lines P^Q^, F^Qi, and P3 $3 parallel 
to the sides of the box and joining points of two opposite faces, 
so that" 

^1 <3i = g-i, A <?2 = I2, P^ Q3 = g.i- 

In a system of progressive waves travelling in the direction determined 
^1 C-iy t^2> f*s ^^^ difference of phase between Pj and Q^ is measured 
by a distance fi^q^, that between P^ and Q^ by f/jfe" and that be- 
tween Pg and Qg by fig^g. The condition for fij, ^^, ,«g, X amounts to 
this') that each of these three lengths must be a multiple of ^X. 
Therefore, if we put 

^'-^/,, '^I'^-^lC,, ^=^g, (150) 

\, h.,, /Cg must be whole positive numbers 
On account of the relation 

f*l +/^| + f^3= Ij 

we have 

and so we now see that for any three whole numbers \, h^, J:.^ there is 
a correspouding set of standing waves. The wave-length is given 
by (151) and the direction constants of the normals to the progres- 
sive waves which we have to combine, by (149) and (150). As these 
progressive waves can have two different states of polarization 2), each 
set of numbers l\, \, \ will lead us to hvo fundamental modes of 
vibration of the ether in the rectangular box, and the energy corre- 
sponding to each set {\, Z^, /,g) will be not ^aT, but ■'^aT^ 

1) Note 40. 2) Note 41. 



JEANS'S THEORY OF RADIATION. 95 

Now, the object of our enquiry is the amount of energy of the 
ether in so far as it belongs to vibrations whose wave-length lies 
between given limits I and k + dX. This amount is 

v-iaT, 

if V is the number of sets of positive integers /,^, l-^, l-^ for which 
the value of A given by (151) lies between I and X + dX. 

74. The number v can easily be calculated if we confine our- 
selves, as we obviously may do, to wave-lengths that are very small 
in comparison with the dimensions g,, q^, q^ of the box. 

Let us consider Iq, \, /cj as the rectangular coordinates of a 
point. Then (151) is the equation of an ellipsoid having for its 
semi-axes 

^, ^S ^- (162) 

Changing X into X ^ dX we get a second ellipsoid, and v will be the 
number of points (/ij. A',, k.^) lying between these two surfaces, whose 
corresponding semi-axes differ by 

(153) 

On account of our assumption concerning the wave-lengths, the 
expressions (152) are very high numbers, and we may even suppose 
that, notwithstanding the smallness of dX, the numbers (153) are 
also very large. This means that all dimensions, the thickness in- 
cluded, of the ellipsoidal shell are very large in comparison with 
the unit of length. 

The number of points with coordinates represented by whole 
numbers, which lie in a part of space whose dimensions are much 
larger than the unit of length, may be taken to be equal to the 
number representing the volume of that part. Remembering that we 
are only concerned with positive values of /i\, 1^2, h^, we find that v 
is equal to the eighth part of the numerical value of the volume of 
the ellipsoidal shell. We have therefore 

and for the energy which we were to calculate 

v.-aT= -3p- dX. 

This is the energy contained in the volume of our rectangular box. 
Dividing by q^q^Ciz, one finds for the energy of radiation in the ether, 



22i 


dX 


2 2, 


dX 


23, 


dX 


X 


X > 


X 


X > 


X 


X 



96 n. EMISSION AND ABSORPTION OF HEAT. 

per unit of volume, so far as it is due to vibrations whose wave- 
length lies between X and A + ^^, 

a result agreeing exactly with (148). 

75. The theory of radiation that was given in §§ 60 — 68 is 
restricted to systems containing free electrons and to the case of very 
long waves. It therefore requires a further development with regard 
to bodies, such as a piece of glass, in which we can hardly admit 
the existence of freely moving electrons, and with regard to the 
shorter waves. If we admit the laws of Boltzmann and Wien, and 
if we take for granted that a curve like that of Fig. 2 represents a 
state of radiation that can be in equilibrium with a ponderable body 
of a given temperature, we must try to account for the form of the 
curve and to discover the ground for the constancy of the product 
X^T. If we succeed in this, we may hope to find in what manner 
the value of this constant is determined by some numerical quantity 
that is the same for all ponderable bodies. 

The theory of these phenomena takes a very diiferent aspect if 
we regard the law of the equipartition of energy as a rule to which 
there is no exception, considering at the same time the ether as a 
continuous medium without molecular structure. Just like any other 
continuous distribution of matter, like a homogeneous string for 
example, a finite part of the ether must then be said to have au in- 
finite number of degrees of freedom; there will be no upper limit to 
the frequency of the modes of vibration that can exist in the ether 
enclosed in the rectangular box of which we have spoken. 

On the contrary, the number of degrees of freedom of a pon- 
derable body is certainly finite if the ultimate particles of which it 
consists are considered as rigid. Consequently, as Jeans has ob- 
served, the theorem of equipartition requires that in a system com- 
posed of a ponderable body and ether, however large be the part of 
space that is occupied by the body, no appreciable part of the total 
energy shall be found in the latter when the equilibrium is reached. 
Indeed, according to Jeans's theory, the formula (148) must be true 
for (M wave-lengths, so that, for a given temperature, we shall find 
an infinite value if, for the calculation of the total amount of energy, 
the expression is integrated as far down as A = 0. This means that, 
if the ether receives any finite amount of energy, such as that which 
is stored up in a body of finite size, the temperature of the ether 
cannot perceptibly rise, the energy being wasted, so to say, for the 
production of extremely short electromagnetic ripples. 



JEANS'S THEORY OF RADIATION. 97 

In order to reconcile these results with observed facts, Jeans 
points out that the emission of rays whose wave lengths are below a 
certain limit may be a very slow process, so slow that a true equi- 
librium is never realized in our experiments. Under these circum- 
stances it is conceivable that, though in length of time all energy of 
a body will be frittered away, yet a certain state may be reached in 
which there are no observable changes, and in which therefore there 
is a kind of spurious equilibrium. 

76. Jeans's conclusions are certainly very important and deserve 
careful consideration. One can imagine three ways in which one 
might escape from them. In the first place, one could suppose the 
number of degrees of freedom of a ponderable body to be itself in- 
finite, either on account of the deformability of the ultimate particles 
or on account of the ether the body contains; this, however, would 
lead us to a contradiction with experiments, because it would require 
a value of the specific heat, far surpassing that to which we are led 
if we attend only to the progressive motion of the molecules. In 
the second place, we could imagine a structure of the ether which 
would make a finite portion of it have only a finite number of de- 
gress of freedom. Lastly, we could altogether abandon the theorem 
of equipartition as a general law. Then, however, we shall be obliged 
to explain why it holds for the case of sufficiently long waves. 

Questions of equal importance and no less difficulty arise when 
we adhere to Jeans's views. It is difficult to believe that, in 
establishing the laws of Boltzmann and Wien, which have been 
so beautifully confirmed by experiment, physicists have been on a 
wholly wrong track. It will therefore be necessary to show for what 
reason those spurious states of equilibrium of which I have spoken 
are subjected to the laws of thermodynamics, and we shall again 
have to find the physical meaning of the constant value of A^T. 

I shall conclude by observing that the law of equipartition which, 
for systems of molecules, can be deduced from the principles of sta- 
tistical mechanics, cannot as yet be considered to have been proved 
for systems containing ether.') 



1) Note 42. 



Lorentz, Theory of electrons. 



CHAPTER III. 

THEORY OF THE ZEEMAN-EFFECT. 

77. The phenomenon of the magnetic rotation of the plane of 
polarization, discovered hy Faraday in 1845, was the first proof of 
the intimate connexion between optical and electromagnetic pheno- 
mena. For a long time it remained the only instance of an optical 
eifect brought about by a magnetic field. In 1877, however, Kerr 
showed that the state of polarization of the rays reflected by an iron 
mirror is altered by a magnetization of the metal, and in 1896 
Zeeman'^) detected an influence of a magnetic field on the emission 
of light. If a source of light, giving one or more sharp lines in the 
spectrum, is placed between the poles of a powerful electromagnet, 
each line is split into a certain number of components, whose distances 
are determined by the intensity of the external magnetic force. 

In my discussion of these magneto-optical phenomena (in which, 
however, I shall not speak of the theory of the Kerr- effect), I shall 
first take the simplest of them all. This is the Zee man- effect, as 
it showed itself in the first experiments, a division of the original 
spectral line into three or two components, the number depending 
on the direction in which the rays are emitted. 

78. I shall first present to you the elementary explanation 
which this decomposition of the lines finds in the theory of electrons, 
and by which it has even been possible to predict certain peculiarities 
of the phenomenon. 

We know already that, according to modern views, the emission 
of light is due to vibratory motions of electric charges contained in 
the atoms of ponderable bodies, of a sodium flame, for example, or 
the luminescent gas in a vacuum tube. The distribution of these 

1) P. Zeeman, Over dea invloed eener magnetisatie op den aard van het 
door een stof uitgezonden licht, Zittinggversl. Amsterdam 5 (1896), p. 181, 242 
[translated in Phil. Mag (5) 43 (1897), p. 226]; Doublets and triplets in the 
spectrum produced by external magnetic forces, Phil. Mag. (5) 44 (1897), p. 55, 
255; Measurements concerning radiation phenomena in the mao-netic field 
ibid. 45 (1898), p. 197. " ' 



ELECTRON VIBRATING IN A MAGNETIC FIELD. 99 

charges and their vibrations may be very complicated, but, if we wish 
only to explain the production of a single spectral line, we can con- 
tent ourselves with a very simple hypothesis. Let each atom (or 
molecule) contain one single electron, having a definite position of 
equilibrium, towards which it is drawn back by an "elastic" force, 
as we shall call it, as soon as it has been displaced by one cause 
or another. Let us further suppose this elastic force, which must 
be considered to be exerted by the other pai'ticles in the atom, but 
about whose nature we are very much in the dark, to be pro- 
portional to the displacement. According to this hypothesis, which 
is necessary in order to get simple harmonic vibrations, the compo- 
nents of the elastic force which is called into play by a displacement 
from the position of equilibrium, whose components are |, tj, t,, may 
be represented by 

-n, -h, -n, 

where /" is a positive constant, determined by the properties of 
the atom. 

If m is the mass of the movable electron, we shall have the 
equations of motion 

whose general solution is 

^ = a cos{n^t + p), t] = a cos (n^t+p), 

t = a" COB in,t + p"), (154) 

a, a, a", p, //, p" being arbitrary constants, and the frequency n^ 
of the vibrations being determined by 



%=Vi- (155) 



Let us next consider the influence of an external magnetic field H. 
This introduces a force given by 

-J[v-H], (156) 

m which expression e denotes the charge of the electron and V its 
velocity. If the magnetic force H is parallel to the axis of s, the 
components of (156) are 

^c 'dt ' e dt ' 

Hence the equations of motion become 

«,i'i=_f£ + ^^^, (157) 

" dt' '^^ e dt' ^ ^ 

7* 



100 III. THEORY OF THE ZEEMAN-EFi'BCT. 



m 



m 



^ = _/-«_«-"-^, (158) 

df I '' c dt > ^ ^ 

dt^ 



= -a. (159) 



79. The last equation shows that the vibrations in the direction 
of 02^ are not affected by the magnetic iield, a result that was to 
be expected, because the force (156) is 0, if the direction of V coin- 
cides with that of H. The particular solution (154) therefore still 
holds. As to the pair of equations (157) and (158), these admit of 
two particular solutions, represented by the formulae 

I = % cos(Wi< + 29i), ri= — a^s\Ta.{n^t + Pi) (160) 

and 

l = a.^Q.os{n^i-\-Pi), ri = a^^Bm{n^t-\-p^), (161) 

in which the frequencies n^ and n^ are determined by 

" '»,n^^n^' (162) 



inc 



and 



v + S'*^ = ^o^ (163) 

whereas a^, a^, p^ and ^^ are arbitrary constants. 

Combination of (154), (160) and (161) gives a solution that 
contains six constants and is therefore the general solution. 

The two solutions (160) and (161) represent circular vibrations 
in a plane perpendicular to the magnetic field, and taking place in 
opposite directions. The frequency n^ of one is higher (if eH, is 
positive) and that of the other lower than the original frequency Wq. 
The j)ossibility of these circular motions can also be understood by 
a very simple reasoning. If the electron describes a circle with 
radius r in a plane perpendicular to H^, and in a direction opposite 
to that which corresponds to this force, there wiL. be, in addition 
to the elastic force fr, an electromagnetic force 

e| v|H, 
c 

directed towards the centre. Both forces being constant, the circular 
orbit can really be described, and we have, by the well known law 
of centripetal force. 



or, since | v| = Mr, 



J, , e I vl H. wv- 
fr + ^-i - = — , 



/.,<■" H, , 

/ + - = mn'', 



from which (162) immediately follows. The equation (163) can be 
found in exactly the same way. 



NATURE OF THE EMITTED LIGHT, 101 

In aU real cases the change in the frequency is found to be 
very smaU in comparison with the frequency itself This shows that, 

even in the most powerful fields, ^ is very small in comparison 

with iif,. Consequently, (162) and (163) may be replaced by 

«i = % + ,' "% "., = % - o"^ ■ ( 1 64) 

The points in the spectrum corresponding to these frequencies lie at 
equal small distances to the right and to the left of the original 
spectral line n^. 

80. We have next to consider the nature of the light emitted 
by the vibrating electron. The total radiation is made up of several 
parts, corresponding to the particular solutions we have obtained, and 
which we shall examine separately. 

Our former discussion (§§ 39 — 41) of the radiation by an electron 
shows that, if such a particle has a vibration about a point 0, along 
a straight line L, the dielectric displacement at a distant point P 
has a direction perpendicular to OP, in the plane LOP, and that, 
for a given distance OP, its amplitude is proportional to the sine 
of the angle LOP. The radiation wiU be zero along the line of vibra- 
tion L, and of greatest intensity in lines perpendicular to it; mo- 
reover, along each line drawn from 0, the light will be plane 
polarized. 

As to a circular vibration, such as is represented by the for- 
mulae (160), its effect is the resultant of those which are produced 
by the two rectiUnear vibrations along OX and OY, into which it 
can be decomposed. We need only consider the state produced 
either in the plane of this motion, or along a line passing through 
the centre, at right angles to the plane. At a distant point P of the 
plane, the light received from the revolving electron is plane pola- 
rized, the electric vibrations being perpendicular to OP, in the plane 
of the circle; if, for example, P is situated on OY, the vibration 
along this line wiU have no effect, and we shall only have the field 
produced by the motion along OX. 

Both components of (160) are, however, effective in producing 
a field at a point on the axis of the circle, i. e. on OZ, the first 
component giving rise to an electric vibration parallel to OX, and 
the second to one in the direction oi OY. It is immediately seen 
that between these vibrations there is exactly the same difference 
of phase as between the two components of (160) themselves, i. e. 
a difference of a quarter period, and that their amplitudes are equal. 
The light emitted along OZ is therefore circularly polarized, the 
direction of the dielectric displacement rotating in the sense corre- 



102 HI- THEORY OF THE ZEEMAN-EFPECT. 

spending to tlie circular motion of the electron. The formulae (160) 
show that, for an observer placed on the positive axis of s, the 
rotation of the electron takes place in the same direction as that of 
the hands of a clock. From this it may be inferred that the rays 
emitted along the positive axis by the motion (160) have a right- 
handed circular polarization. 

Similar considerations apply to the motion represented by (161). 
The radiation issuing from it in the direction just stated has a left- 
handed circular polarization. If it is further taken into account that 
the frequency of the rays is in every case equal to that of the motion 
originating them, one can dravt^ the following conclusions, which have 
been fully verified by Zeeman's experiments.^) 

Let the source of light be placed in a magnetic field whose 
lines of force are horizontal, and let the light emitted in a horizontal 
direction at right angles to the lines of force be examined by means 
of a spectroscope or a grating. Then we shall see a triplet of lines, 
whose middle component occupies the place of the original line. 
Each component is produced by plane polarized light, the electric 
vibrations being horizontal for the middle line, and vertical for the 
two outer ones. 

If, however, by using an electromagnet, one core of which has 
a suitable axial hole, we examine the light that is radiated along the 
lines of force, we shall observe only a doublet, corresponding in po- 
sition to the outer lines of the triplet. Its components are both 
produced by circularly polarized light, the polarization being right- 
handed for one, and left-handed for the other. 

81. After having verified aU this, Zeeman was able to obtain 
two very remarkable results. In the first place, it was found that, 
for light emitted in a direction coinciding with that of the magnetic 
force, i. e., if H^ is positive, in that of OZ, the polarization of the 
component of the doublet for which the frequency is lowest, is right- 
handed This proves that, for a positive value of H^, the first of 
the two frequencies given by (1G4) is the smaller. Therefore, the 
charge e of the electron to whose motion the radiation has been 
ascribed must be negative. This agrees with the general result of 
other lines of research, that the negative charges have a greater mo- 
bility than the positive ones. 

The other result relates to the ratio between the numerical values 
of the electric charge and the mass of the movable electrons. This 
ratio can be calculated by means of the formulae (164), as soon as 
the distance between the components, from which we can deduce 

1) Note 43. 



SERIES OF SPECTRAL LINES. 103 

Wj — Mq, and tlie strength of the magnetic field have been measured. 
The number deduced by Zeeman from the distance between the 
components of the D- lines, or rather from the broadening of these 
lines, whose components partly overlapped each other, was one of 

the first values of — that have been published. In order of magni- 
tude it agTees with the numbers that have been found for the nega- 
tive electrons of the cathode-rays and the /3-rays. 

Unfortunately, the satisfaction caused by this success of the 
theoi-y of electrons in explaining the new phenomenon, could not last 
long. It was soon found that many spectral lines are decomposed 
into more than three components, four, six or even more^), and till 
the present day, these more complicated forms of the Zeeman-effect 
cannot be said to have been satisfactorily accounted for. 

All I can do, will therefore be to make some suggestions as to 
the direction in which an explanation may perhaps be looked for. 

82. Before proceeding to do so, I may be permitted briefly to 
mention some of the important results that have been found in the 
examination of the distribution of spectral lines, such as they are in 
the absence of a magnetic field. In the spectra of many elements 
the lines arrange themselves in series, in such a manner that, for 
each series, the fi-equencies of all the lines belonging to it can be 
represented by a single mathematical formula. The first formula of 
this kind was given by Balmer^) for the spectrum of hydrogen. After 
him, equations for other spectra have been established by many 
physicists, especially by Rydberg^) and by Kayser and Runge.*) 

For our purpose it will be sufficient to mention some examples. 

In the spectrum of sodium three series of double lines have 
been found, which are distinguished by the names of principal series, 
first subordinate or nebulous series, and second subordinate or sharp 
series. We may also say that each of the three is composed of two 
series of single lines, one containing the less refrangible, and the 
other the more refrangible lines of the doublets. 



1) In later researches a decompositioD into no less than 17 components 
has been observed. 

2) J. J. Balmer, Notiz uber die Spektrallinien des Wasserstoffs, Ann. Phys. 
Chem. 26 (1885), p. 80. 

3) J. R. Rydberg, Recherches sur la constitution des spectres d'emission 
des elements chimiques, Svenska Vetensk. Akad. Handl. 23 (1889), No. 11; La 
distribution des raies spectrales, Rapports pres. au Congrfes de physique, 1900, 
2, p. 200. 

4) H. Kayser u. C. Runge, tjber die Spektren der Alkalien, Ann. Phys. 
Chem. 41 (1890), p. 302; Uber die Spektra der Elemente der zweiten Mende- 
lejefr'schen Gruppe, ibid. 43 (1891), p. 385 



104 IH- THEORY OF THE ZEEMAN-EFPECT. 

The frequency in these six series, measured by the number n 
of wave-lengths in a centimetre, has been represented by Rydberg 
by means of the formulae contained in the following table. 

Principal series I A = ^-^-, - ^--A^-, , (165) 

11 -- = - ~ - (166) 

First subordinate (nebulous) series I ^ = jyt. — \i~\„^ArsY> (167) 

" " " " ^^ W,^ (1 + (x,)* ~ l^-\rsy ' *-^^^) 

Second subordinate (sharp) series I ^ = tt— r- — r^ — -, — xT-Ya ' 0-^^) 

" " " " iV„ (l + fi-s,)' {m + ay ^ ' 

In these equations, N^, ftj, ^f.^, 6 and d are constants having the 
values 

N„ = 109675 

iUi = 1,1171, ft2 = 1,1163, ^ = 0,9884, (? = 0,6498 

and we shall find the frequencies of the successive lines in each 
series by substituting for ni successive positive whole numbers. If, 
in doing so, we get for n a negative value — n, this is to mean 
that there is a line of the frequency u'. 

83. I particularly wish to draw your attention to the following 
remarkable facts that are embodied in the above formulae. 

1. If the value of m is made continually to increase, that of ii 
increases at the same time, converging however towards a finite 
limit, corresponding to -m = oo, and given for the different series by 

^. ^0 etc 

ii + cr' (i+ft)^' ®'°-_ 

The lines of a series are not placed at equal distances from each 
other; as we proceed towards the side of the ultra-violet, the lines 
become crowded together, the series being unable, so to say, to 
pass the limiting position of the line given by one of the above 
numbers. 

As to the number of lines that have been observed, this varies 
from one series to the other. If the above formulae (or equations 
of a similar kind) are the expression of the real state of things, the 
number of lines is to be considered as infinitely great. 

2. The frequencies of a doublet of the first subordinate series (I, II) 
are obtained, if in (167) and (168) we substitute for jw the same 



CONNEXION BETWEEN DIFFERENT SERIES. 105 

mimber. These frequencies differ by 

-^0 N„ _ 

whatever be the value of m. The same difference is found, if we 
calculate the frequencies of a doublet of the second subordinate 
series (I, II). Therefore, if the distance between two lines is measured 
by the difference of their frequencies, the interval between the two 
components is the same for aU the doublets of the first and of the 
second subordinate series. 

It is otherwise with the doublets of the principal series (I, II). 
The distance between the two components is given by 



■A quantity, which diminishes when m increases, and approaches the 
limit for m = oo. 

In connexion with this, it must be noticed that the convergence 

frequency has the same value 77-1° ,, for the members I and II of 
the principal series. 

3. This is not the only connexion between different series. 

The formulae show that the convergence frequencies are >, -.-"—.r, and 

7 , " ,; , both for the first and the second subordinate series (I, II). 

Finally, it is important to remark that, if in (165) and (166) we 
put »« = 1, we get the same frequencies as from (169) and (170) 
for the same value of m. The doublet with these frequencies can 
therefore be considered to be at the same time the first of the prin- 
cipal, and the first of the second subordinate series. 

We may further say that the entire principal series I and the 
entire sharp series I correspond to each other, being both charac- 
terized by the constants fij and 0, and that there is a similar relation 
between the principal series II and the sharp series II. In this 
connexion it is proper to remark that the more refrangible lines of 
the principal doublets correspond to the less refrangible ones of the 
sharp doublets, and conversely. If, for example, fi^ is greater than [i^, 
the first constant will give the larger frequency in the principal 
series, and the lesser frequency in the second subordinate one. 

4. Similar results have been obtained for the other alkali metals, 
which also show series of doublets in their spectrum, and for magne- 
sium, calcium, strontium, zinc, cadmium and mercury. Only, in the 
spectra of these latter metals, one finds series, not of doublets but of 
triplets. To the scheme given in the formulae (165) — (ITO), we have 



106 HI. THEORY OF THE ZBEMAN-EFFECT. 

therefore to add in this case: 

Principal series III J = ^^-±^, _ ^-^^^^1^ , 

First subordinate series III ^ = tt-j^ — y — v . g y. , 

Second ,, ,, III -rr = 



^0 (l+f's)' (m+ey 

However, even thus the scheme is not yet complete. In the spectrum 
of mercury, for example, there is a certain number of additional 
lines, which closely accompany those of which we have just spoken, 
and which are therefore often called satellites. 

These again show certain remarkable regularities. They occur 
in the first subordinate series (I, II, III), but not in the second sub- 
ordinate one. In each triplet of the first series, there are three satellites 
accompanying the first line of the triplet, two belonging to the 
second, and one for the third, so that the triplet is really a group 
of nine lines. 

As to the principal series of the last named elements, I have 
added them only for the sake of analogy. Principal series of triplets 
have not yet been observed. 

84. It is only for a comparatively small number of chemical 
elements, that one has been able to resolve the system of their 
spectral lines, or at least the larger part of them, into series of the 
kind we have been considering. In the spectra of such elements as 
gold, copper and iron, some isolated series have been discovered, but 
the majority of their lines have not yet been disentangled. Never- 
theless, it cannot be denied that we have made a fair start towards 
the understanding of line spectra, which at first sight present a 
bewildering confusion. There can be no doubt that the lines of a 
series really belong together, originating in some common cause, and 
that even different series must be produced by motions between which 
there is a great resemblance. 

The similarity of structure in the spectra of elements that re- 
semble each other in their chemical properties, is also very striking. 
The metals in whose spectra the lines are combined in pairs are all 
monovalent, whereas the above series of triplets belong to divalent 
elements. Perhaps the most remarkable of all is the fact, that Rydberg 
was able to represent aU series, whatever be the element to which 
they belong, by means of formulae containing the same number N^. 
This equality, rigorous or approximate, of a constant occurring in 
the formulae of the different elements, must of course be due to 
some corresponding equality in the properties of the ultimate par- 



DIVISION OP THE LINES OF A SERIES. 



107 



tides of which these elements consist, but at present we are whoUy 
unable to form an idea of the nature of this similarity, or of the 

physical meaning of the length of time corresponding to ^ -' • 

-"0 

85. The investigation of the Zeeman-effect for a large number 
of spectral lines, to which many physicists have devoted themselves 
of late years, has fully confirmed the hypothesis of an intimate con- 
nexion between the different spectral lines of a substance; it has 
furnished rich material for future research, but which, in the present 
state of theory, we can understand only very imperfectly. 

Before saying a few words of the results that have been obtained, 
I must revert once more to the elementary theory of the triplets and 
to the formulae (164) we deduced from it. These show that, if aU 
spectral lines were split according to the elementary theory, and if, 

in all cases, the ratio — had the same value, we should always ob- 
serve triplets with the same difPerence of frequency between their 
components. This is what, for the sake of brevity, I shall call an 
equal splitting of the lines. 

Now, the measurements of Runge and Paschen^) and other 
physicists have led to a very remarkable result. Though there are 
a large number of spectral lines which 
are split into more than three components, 
and though even the triplets that have 
been observed, are not equal to each 
other in the above sense, yet all lines 
forming a series, i. e. all lines that can 
be represented by one and the same 
formula, are divided in exactly the same 
way, and to exactly the same extent. 
There seems to be no doubt as to the 
validity of this general law. 

In those series which consist of triplets or doublets, the mode 
of division of the lines is in general different for the lines of one 
and the same triplet or pair, but, according to the law just mentioned, 
the same mode of division repeats itself in every triplet or every 
doublet. Thus, in each triplet belonging to the second subordinate 



n n n ;a 



7% TV n. 





















n, 7- 


!. p p n n. 
















n P rt 











I'ig. 3. 



1) Several authors have tried to establish formulae by which the distribu- 
tion of the lines of a series can be represented still more accurately than by 
those of Rydberg. See, for instance, W. Ritz, Ann. Phys. 12 (1903), p 264, 
and E. E. Mogendorff, Amsterdam Proc. 9 (1906), p. 434. 

2) C. Runge, tjber den Zeeman-Effekt der Serienlinien, Phys. Zeitschr. 3 
(1902), p. 441; C. Runge u. F. Paschen, Uber die Strahlung des Quecksilbers 
im magnetischen Felde, Anhang z. d. Abhandl. Akad. Berlin, 1902, p. 1. 



7> 


TV 


P » 


n 


71, 














n p p n 













108 III. THEORY OF THE ZEEMAN-BFFECT. 

series of mercury, the less refrangible line is split into nine com- 
ponents, the middle line into six, and the most refrangible line into 
three components. These divisions are shown in Fig. 3, in which the 
letters |) and n mean that the electric vibrations of the line are 
parallel or perpendicular to the lines of force. 

Equal modes of division are found not only in the different 
lines of one and the same series, but also in the corresponding series 

of different elements. For example, the 
lines Z'j andDg of sodium, which form 
the first member of the principal series, 
are changed into a quartet (Cornu's 
quartet) and a sextet (Fig- 4), and the 
first terms in the principal series of 
copper and silver present exactly the 
rig. 4. same division. 

86. Vou see from all this that the phenomena are highly com- 
plicated, and that there would be a bewildering intricacy, were it not 
for the law which I have just pointed out to you, which reveals 
itself in the decomposition of the lines of the same series, or of 
corresponding series. Nor is this the only case in which a connexion 
has been found between the Ze em an- effect for different lines. Fig. 3 
shows another most remarkable regularity. The distances represented 
in it can aU. be considered as multiples of one number, and the 
same can be said of many of the displacements that have been observed 
by Runge and Paschen in the spectrum of mercury. A similar 
remark applies to the case of Fig. 4.^) 

I should also mention that the interesting connexion between 
the principal series and the second subordinate series of which we 
have already spoken, is beautifully corroborated by the observations 
of the Zeeman-effect. The more refrangible components of the doublets 
of one of these series are split in the same way as the less refrangible 
components of the doublets of the other. 

Finally, it must not be forgotten that, although a very large 
number of lines show a rather complicated Zeeman-effect, especially 
those which belong to the series of which we have spoken, yet there 
are also many lines which are changed into triplets by the action of 
a magnetic field. In the recent work of Purvis, for example, no 
less than fifty cases of this kind have been found in the spectrum of 
palladium. I must add that many more lines of this element are 
decomposed in a different way. 

1) See on this question of the oommensurability of the magnetic separations 
in different cases, C. Runge, tjber die Zerlegung von Spektrallinien im mao-- 
netischen Felde, Phys. Zeitschr 8 (1907), p. 232. ° 



SLNUFFIGIENCT OF THE ELEMENTARY THEORY, 109 

87. It has already been mentioned that Zeeman's first deter- 
mination of the ratio — led to a value of the same order of maffni- 

tude as that which has been found for the electrons of the cathode 
rays and the /3-rays of radium. Later measurements have shown, 
however, that the distance between the components is not the same 

in different triplets, and that therefore different values of " are found, 

m ' 

if the formulae (164) are applied in aU cases. Though some triplets give 

a value of — equal to the number found for free negative electrons, 

the result is different in the majority of cases. This can be attributed, 

either to real differences between the values of -, or to the imper- 

fectness of the elementary theory. I believe that there is much to be 
said in favour of the latter alternative. After all that has been said, 
we cannot have much confidence in the formulae (164), but there are 
strong reasons for believing in the identity of aU negative electrons. 

88. If time permitted it, it would be highly interesting to con- 
sider some of the hypotheses that have been put forward in order 
to explain the structure of spectra and the more complicated forms 
of the Zeeman -effect. There can be no difference of opinion as to 
the importance of the problem, nor, I believe, as to the direction 
in which we have to look for a solution. The liability of spectral 
lines to be changed by magnetic influences undoubtedly shows, what 
we had already assumed on other grounds, that the radiation of Kght 
is an electromagnetic phenomenon due to a motion of electricity in 
the luminous particles, and our aim must be to explain the observed 
phenomena by suitable assumptions concerning the distribution of the 
charges and the forces by which their vibrations are determined. 

Unfortunately, though many ingenious hypotheses about the struc- 
ture of radiating particles have been proposed, we are still very far 
from a satisfactory solution. I must therefore confine myself to some 
general considerations on the theory of the Zeeman-effect, and to 
the working out of a single example which may serve to illustrate 
them. 

89. In the first place, we can leave our original hypothesis of 
a single movable electron for a more general assumption concerning 
the structure and properties of the radiating particles. Let each of 
these be a material system capable of very small vibrations about a 
position of stable equilibrium, and let its configuration be determined 
by a certain number of generalized coordinates Pi, p^, ■ ■ ■, P^r We 
shall suppose these to be chosen in such a manner, that they are 



110 III. THEORY OF THE ZEBMAN-EFFECT. 

in the position of equilibrium, and that the potential energy and 
the kinetic energy are represented by expressions of the form 

Then, Lagrange's equations of motion become 

Wift = - fiPi > ^ih = -UPi, ■■■! ^f,i^, = - f^Pfr (I'^i) 
Since each of these formulae contains but one coordinate, the changes 
of one coordinate are wholly independent of those of the other, so 
that each equation determines one of the fundamental modes of 
vibration of the system. The frequencies of these modes, and the 
positions of the corresponding spectral lines are given by 

We shall now introduce an external magnetic force il, which of 
course may be considered to be the same in all parts of our small 
material system. In order to make this force have an influence on 
the vibrations, we shall suppose the parts of the system to carry 
electric charges, which are rigidly attached to them, so that the 
position of the charges is determined by the coordinates p. 

As soon as the system is vibrating, the charges are subjected 
to forces due to the external magnetic field. These actions can be 
mathematically described by the introduction into the equations of 
motion of certain forces in the generalized sense of the word. Deno- 
ting these forces by Pi, Pg, . . ., P , we shall have, instead of (171), 

»%Pi =-fiPi + ^i, etc. 
Without a knowledge of the structure of the vibrating system, 
and of the distribution of its charges, it is of course impossible, 
completely to determine Pj, Pg, .... One can show, however, that 
the expressions for these quantities must be of the form 

A = Cisft + c^ih + •■■ + %Pto 

P2 = c^iPi + ^2sh -i + %Pf„ ' (173) 

etc., 

where the constants c axe proportional to the intensity of the magnetic 
field.^) Between these coefficients there are the following relations 

Csi = — C12, C32 = — C23, etc. (174) 

The proof of all this is very easy, if we remember the funda- 
mental expression — [v • h] for the action of a field on a moving 

1) Note U. 



VIBRATIONS OF A CHARGED SYSTEM IN A MAGNETIC FIELD. lU 

charge. The components of this action along the axes of coordinates 
are linear and homogeneous functions of the components of the velo- 
city V. Consequently, all the rectangular components of the forces 
acting on the vibrating particle must be functions of this kind of 
Pi} ih> • • •! P^ii because the velocity of any point of the system is a 
linear and homogeneous function of these quantities. The same must 
be true of the Lagrangian forces Pj, P^, . . ., P , because these are 
linear and homogeneous functions of the rectangular components of 
the forces. 

In order to find the relations between the coefficients c, we have 
only to observe that the work of the additional foi-ces Pj, P^, etc. 
is 0, because the force exerted by the magnetic field on a moving 
charge is always perpendicular to the line of motion. The condition 

P^Pi + P,i>, + --- + P,P, = o, 

to which we are led in this way, is the ground for the relations (174) 
and for the absence of a term with p^ in the first of the equations 
(173), of one with p^ in,' the second, etc. 

90. The equations of motion 

^iPi +fiPi = Ciik + f^iih H ^- %^P^o 

etc. 
can be treated by well known methods. Putting 

l\ = q,^'"\ P2 = q,s"", ■■■, P,u = 9^^"", (175) 

where n, q^, q^, ■ ■ ■, q^^ are constants, we find the (i equations 

(/; - m^if) g-i - inc^^q^ — inc^^q^ ^wCj^.g,, = 0, | 

- ^nc.,^1i. + (/"s— >>«2«^)?2 - *'«C23?3 i'>^%,% ^^'\ ^^'^^^ 

etc. J 

If, from these, the quantities q^, q^, ■ ■ ■ , q^, are eliminated, the result 
is an equation which determines the coefficient n. On account of the 
relations (174), and the smallness of the terms with c^g, qj, etc., it 
may be shown that the equation contains only n^, and that it gives /i. 
real positive values for this latter quantity. Hence, there are ,u. 
positive numbers n^', w/, . . . such that the resulting equation is 
satisfied by 

n = ± Wi', n = ± nj, . . . , n = ±n^. 

For each of these values of n, the ratios between q^, q^, ..., q^ 
can be deduced from (176). Finally, if we take the real parts of 



112 III. THEORY OF THE ZEEMAN-BFFECT. 

the expressions (Hfi), we find (i fundamental modes of vibration, 
whose frequencies are 



It is easily seen from this that, if we do not assume any special 
relations between the constants involved in our problem, there wiU 
be no trace at all of the Ze em an- effect. In the absence of the 
magnetic field we had p spectral lines, corresponding to the frequen- 
cies %, n.2, . . ., w^,. The effect of the field is, to replace these by 
the slightly different values n^, n^, ..., 11^,, so that the lines are 
shifted a little towards one side or another, without being split into 
three or more components.-'^) 

91. The assumption that is required for the explanation of the 
Zeeman-effect can be found without any calculation. Let us imagine, 
for this purpose, a source of light placed in a magnetic field, and 
giving in the spectrum a triplet instead of an original spectral line. 
The components of this triplet are undoubtedly due to three modes 
of motion going on in the interior of the radiating particles, and these 
modes must be different from each other, because otherwise their fre- 
quencies ought to be the same. Let us now diminish the strength of 
the field. By this the components are made to approach each other, 
perhaps so much, that we can no longer distinguish them, but the 
three modes of motion will certainly not cease to be there. Only, their 
frequencies are less different from each other than they were in the 
strong field. By continually weakening the field, we can finally obtain 
the case in which there is no field at all, but even then the three 
modes of motion must exist. They still differ from each other, but 
their frequencies have become equal. 

The necessary condition for the appearance of a magnetic triplet 
is thus seen to be that, in the absence of a magnetic field, thi-ee of 
the frequencies n^ ,n^, . . ., n , corresponding to three different degrees 
of freedom, are equal to each other, or, as I shall say for the sake 
of brevity, that there are three equivalent degrees of freedom. Then, 
the magnetic field, by which all the frequencies are changed a little, 
produces a slight inequality between the three that were originally 
equal. We can express the same thing by saying that only a spectral 
line which consists of three coinciding lines can be changed into a 
triplet, the magnetic field producing no new lines, but only altering 
the positions of already existing ones. 

1) Note 45. 



SHARPNESS OP THE MAGNETIC COMPONENTS. 113 

92. These conclusions, which one can easily extend to quartets, 
quintets etc., are fully corroborated by the mathematical theory. If 
originally 

we shall have, under the influence of a magnetic field, the three 
frequencies 

Ml and Wi±4-l/-^^ + -^^+^^, (177) 

indicating the existence of a symmetrical triplet, the middle Kne of 
which has the position of the original spectral line. In a similar 
manner it can be shown that we shall obsetve a quartet, a quintet, 
etc., whenever the system has four, five or more equivalent degrees 
of freedom. All these more complicated forms of division of a 
spectral line are found to be symmetrical to the right and to the 
left of the original position, so that, if the number of components is 
odd, the middle one always occupies the place of the primitive line.^) 

93. The existence of a certain number of equivalent degrees of 
freedom is not the only condition to which we must subject the 
radiating particles. The fact that the magnetic components of the 
spectral lines have the same degree of sharpness as the original lines 
themselves requires a further hypothesis. We can understand this 
by reverting for a moment to the expression (177). In it, the coef- 
ficients C23, C31, Cj2 are linear and homogeneous functions of the com- 
ponents H^, y\y, H^ of the external magnetic force. Therefore, the 
distance between the outer components of the triplet and the middle 
one is given by an expression of the form 



Van H/ + q,, H/ + ^33 H/ + 22,, H, H^ + 2q,, H^H, + 2q,, H,H,, (178) 

in which q^, q^, ■ ■ ■ , q^, ■ ■ ■ are constants depending on the nature 
of the vibrating particle. K, without changing the direction of the 
field, its intensity is doubled, the distance between the lines will in- 
crease in the same ratio. So far our formula agrees with experi- 
mental results.^) 

Let us now consider the tafluence of a change in the direction 
of the magnetic field, the intensity | H | being kept constant. By 
turning the field we shall give other values to H^, H^^, H., and also 
to the expression (178). It is clear that the same change will be 
brought about if, leaving the field as it is, we turn the radiating 
particle itself. Hence, if the source of light contains a large number 
of particles having all possible orientations, the distance (178) will 



1) Note 46. 2) Note 47. 

Lorentz, Theory of electrons. 



114 III. THEORY OF THE ZEEMAN-EFFECT. 

vary between certain limits, so that the outer lines of the observed 
triplet, which is due to the radiation of all the particles together, 
must be more or less diffuse. 

Since it is difficult to admit that the particles of a luminous 
gas, when subjected to a magnetic field, are kept in one definite 
position, the only way of explaining the triplet with sharp outer com- 
ponents seems to be^) the assumption that the coefficients in (178) 
are such that the quadratic function takes the form 

In this case, the influence of a magnetic field on the frequencies is 
independent of the direction of the force relatively to the particle. 
As regards this influence, the particle can then be termed isotropic. 
The simple mechanism which we imagined in the elementary 
theory of the Zeeman-efFect obviously fulfils the conditions to which 
we have been led in what precedes. Indeed, a single electron which 
can be displaced in all directions from its position of equilibrium, 
and which is pulled back towards this position by a force indepen- 
dent of the direction of the displacement, has the kind of isotropy 
we spoke of just now. It has also three degrees of freedom, corre- 
sponding to the displacements in three directions perpendicular to 
each other. 

94. The question now arises, whether we can imagine other, 
more complicated systems fulfilling the conditions necessary for the 
production of magnetic quartets, quintets etc. In order to give an 
example of a system of this kind, I may mention the way in which 
A. A. Robb^) has explained a quintet. For this purpose he supposes 
that a radiating particle contains two movable electrons, whose posi- 
tions of equilibrium coincide, and which are pulled towards this 
position by elastic forces proportional to the displacements, and deter- 
mined by a coefficient that is the same for both electrons. The 
charges and the masses are also supposed to be equal. Robb does 
not speak of the mutual electric action of the electrons, but he in- 
troduces certain connexions between their positions and their motions. 
If Pj and Pg are vectors drawn from the position of equilibrium 
towards the two electrons, and fjg the vector drawn from the first 
electron towards the second, these connexions are expressed by the 
equations 



1) See, however, Note 64. 

2) A. A. Robb, Beitrage zur Theorie des Zeeman-Effektes, Ann. Pliys. 16 
(1904j, p. 107. 



VIBRATIONS OF CHARGED SPHERICAL SHELLS. 115 

r,,^ = /c(r,^+r,^), 

where fc is a constant. It is immediately seen that in all these 
assumptions there is nothing that relates to a particular direction in 
space. On account of this, the five different frequencies which are 
found to exist under the influence of a magnetic force, are independ- 
ent of the direction of this force, and a large number of systems 
of the kind described would give rise to a quintet of sharp lines. 

Robb has worked out his theory at a much greater length than 
appears from the few words I have said about it, and it certainly is 
very ingenious. Yet, his hypothesis about the connexions between 
the two electrons seems to me so artificial, that I fear he has given 
us but a poor picture of the real state of things. 

95. The same must be said of an hypothesis which I tried many 
years ago. After having made clear to myself that the vibrating par- 
ticles must be isotropic, I examined the motions of systems surely 
possessing this property, namely of uniformly charged spherical shells, 
having an elasticity of one kind or another, and vibrating in a mag- 
netic field. By means of the theory of spherical harmonics, the dif- 
ferent modes of motion corresponding to what we may call the dif- 
ferent tones of the shell, can easily be determined, and it was found 
that each of the tones can originate in several modes of motion, so 
that we can truly say that each spectral line (if the vibrations can 
produce light) consists of a certain number of coinciding lines, this 
number increasing as we pass on to the higher tones of the shell. The 
calculation of the influence of an external magnetic force confirmed 
the inference drawn from the general theory; if a certain frequency 
can be produced in 3, 5 or 7 independent ways, the spectral line 
corresponding to it is split into 3, 5 or 7 components. 

For more than one reason, however, this theory of vibrating 
spherical shells can hardly be considered as anything more than an 
illustration of the general dynamical theorem; it cannot be said to 
furnish us with a satisfactory conception of the process of radiation. 
In the first place, if the series of tones of the shell gave rise to the 
successive members of a series of spectral lines, the number of com- 
ponents into which these are divided in a magnetic field ought to 
increase as we proceed in the series towards the more refrangible 
side. This is in contradiction with the results of later experience, 
vhich has shown, as I already mentioned, that all the lines of a series 
are split in exactly the same way. 



116 ni. THEORY OF THE ZEEMAN-EFFECT. 

In the second place, I pointed out that the spherical shells, when 
vibrating in their higher modes, are very poor radiators. In these 
modes the surface of the shell is divided by nodal lines into parts, 
vibrating in different phases, so that the phases are opposite on both 
sides of a nodal line. The vibrations issuing from these several parts 
must necessarily destroy each other for the larger part by inter- 
ference. 

96. In the light of our present knowledge, a third objection, 
which is a very serious one, may be raised. Though in the Zeeman- 
effect the separation of the components is not exactly what it would 

be, if in the formulae (164) the ratio — had the value that has been 

deduced from experiments on cathode-rays, yet it is at least of the 
same order of magnitude as the value which we should find in this 

case. Hence, if we write ( — ) for the ratio deduced from the obser- 

vations on cathode-rays, and if we use the symbol (=) to indicate 
that two quantities are of the same order of magnitude, we have for 
the distance between two magnetic components the general formula 

On the other hand, the theory of the vibrating shells leads to 
an equation of the form 

•^"Hl;^' (180) 

in which e^ is the charge and m^ the mass of the shell. 
We may infer from (179) and (180) that 

^ (=) (-) 

an equation which shows that the properties of the charged sphere 
cannot be wholly different from those of a free electron. Therefore, 
as we know that the mass m.^ of such an electron is purely electro- 
magnetic, we are led to suppose that the mass m^ of the shell is of 
the same nature. This, however, leads us into a difficulty, when we 
come to consider the frequencies of the vibrations. The relative 
motions of the parts of the shell are in part determined by the electric 
interactions of these parts, and even if they were wholly so, i. e. if 
there were no „elasticity" of an other kind, the wave-lengths cor- 
responding to the different tones as I have called them, would, on 
the above assumption concerning the mass, be extremely small; they 
would be of the same order of magnitude as the radius R^. They 
would be still smaller if there were an additional elasticity. There- 



THOMSON'S MODEL OF AN ATOM. 117 

fore, as the radius B^ must certainly be very much smaller than the 
wave-length of light, we can never hope to explain the radiation of 
light by the distortional vibrations of spheres whose charge and 
radius are such as is required by the magnitude of the Ze em an- effect. 

97. It is clear in what way we can escape from the difficulty 
I just now pointed out. We must ascribe the radiation, not to the 
distortional vibrations of electrons, but to vibrations in which they 
move as a whole over certain small distances. Motions of this kind 
can exist in an atom which contains a certain number of negative 
electrons, arranged in such a manner that they are in stable equili- 
brium under the influence of their mutual forces, and of those that 
are exerted by the positive charges in the atom. This conception is 
very like an assumption that has been developed to a considerable 
extent by J. J. Thomson^), and according to which an atom consists 
of a positive charge uniformly distributed over a spherical space, 
a certain number of negative electrons being embedded in this sphere, 
and arranging themselves in a definite geometrical configuration. 

In what follows, it will be found convenient to restrict the name 
of electrons to these negative particles or, as Thomson calls them, 
„corpuscles". 

If the atom as a whole is uncharged, the total positive charge 
of the sphere must be equal to the sum of the charges of the nega- 
tive electrons; we can, however, also conceive cases in which this 
equality does not exist. 

It is interesting to examine the dimensions that must be ascribed 
to a structure of the above kind. Let the mutual distances of the 
electrons be of the same order of magnitude as a certain line I, and 
let e be the charge of each electron. Then, the repulsion between 

two electrons is of the same order as - — ^, and the change which 

this force undergoes by a very small displacement d of one of the 
corpuscles, is of the order 

This change may be considered as an additional force that is called 
into play by the displacement d. Hence, if we exclude those cases 
in which a very large number of electrons produce additional forces 
of the same direction, and also those in which the additional force 
which is due to the negative electrons is compensated or far surpassed 
by that which is caused by the positive charge, the total force by 



1) J. J. Thomson, The corpuscular theory of matter, London, 1907. 
chap. 6 and 7. 



118 HI. THEORY OF THE ZEEMAN-EFPECT. 

which an electron is pulled back towards its position of equilibrium 
is given, as to order of magnitude, by the above expression. I shall 
suppose the electrons to have the same radius B, charge e and mass m 
as the free negative electrons, and I shall write X for the wave-length 
corresponding to their vibrations. Now, by what precedes, we have 
for the frequency 

n^ (=) -, m y 



or, on account of (72), 

But 

so that 



'^^Hl¥ 



27CC 



H=)p^ 



Putting A = 0,5 ■ 10~*cm, and introducing the value of B (§ 35), one 
finds by this equation 

I (=) 2,4 • 10-8 cm (_^ 1^6 . 105 jj. 

This means that the electrons must be placed at distances from each 
other that are very much larger than their dimensions, so that, com- 
pared with the separate electrons, the atom is of a very large size. 
Nevertheless, it is very small compared with the wave-length, for 
according to the above data we have 

;(=)5-l0-n. 

One consequence of the high value which we have found for I : B is 
that the electromagnetic fields of the electrons do not appreciably 
overlap. This is an important circumstance, because, on account of '■ 
it, we may ascribe to each electron the electromagnetic mass m which 
it would have if it were wholly free. 

The value we have found for I is of about the same order of 
magnitude as the estimates that have been formed of molecular 
dimensions. We may therefore hope not to be on a wrong track if, 
in the above manner, we try to explain the production of light by 
the vibrations of electrons under the influence of electric forces. 

98. It is easily seen that a number of negative electrons can 
never form a permanent system, if not held together by some ex- 
ternal action. This action is provided for in J. J. Thomson's model 
by the positive sphere, which attracts all the electrons towards its 
centre 0, and which must be supposed to extend beyond the electrons, 
because otherwise there could be no true static equilibrium. As 



NEGATIVE ELECTRONS IN POSITIVE SPHERE. 119 

already stated, I shall use the same assumption, but I shall so far 
depart from Thomson's ideas as to consider the density q, not as 
constant throughout the sphere, but as some unknown function of 
the distance r from its centre. The greater generality that is obtained 
in this way will be seen to be of some interest. With a slight modi- 
fication, our formulae might even be adapted to the case of electrons 
attracted towards the point by some force /"(r) of unknown origin, 
for any field of force that is symmetrical around a centre 0, can be 
imitated by the electric field within a sphere in which the density p 
is a suitable function of r. 

However, I shall suppose q to be positive in all layers of the 
sphere, and to decrease from the centre outward. 

As is well known, the general outcome of the researches on the 
a-rays of radio-active bodies and on the canal rays has been that 
the positive electricity is always attached to the mass of an atom.*) 
In accordance with this result, we shall consider the positive sphere 
as having nearly the whole mass of the atom, a mass that is so 
large in comparison with that of the negative electrons, that the 
sphere can be regarded as immovable, while the electrons can be dis- 
placed within it. The question as to whether the mass of the posi- 
tive sphere is material or electromagnetic, can be left aside. Of course 
the latter alternative must be discarded, if we apply to the positive 
electricity a formula similar to the one we have formerly given for 
the electromagnetic mass of an electron; on account of the large 
radius of the sphere, the mass calculated by the formula would be 
an insignificant fraction of the mass of the negative electrons. It 
might however be that part of the charge is concentrated in a large 
number of small particles whose mutual distances are invariable; in 
this case the total electromagnetic mass of the positive charge could 
have a considerable value. 

99. Before passing on to a special case, some other remarks may 
be introduced. 

In the first place, an atom which contains N movable negative 
electrons, wiU have 3N degrees of freedom. Consequently, if its 
vibrations are to be made accountable for the production of one or 
more series of spectral lines, the number of electrons must be rather 
large. It ought even to be infinite, if a series really consisted of an 
infinite number of lines, as it would according to Rydberg's equa- 
tions. Since, however, these formulae are only approximations, and 
since the lines that can actually be observed are in finite number, 
I believe this consideration need not withhold us from ascribing the 



1) See, however, Note 64. 



120 in. THEORY OF THE ZEEMAN-EPEEGT. 

radiation of light to atoms containing a finite, thougli perhaps a rather 
large number of negative electrons. 

In the second place we shall introduce the condition that the 
vibrating system must be isotropic. True isotropy, i. e. perfect 
equality of properties in all directions, can never be attained by 
a finite number of separated particles. It is only when we are con- 
tent with the explanation of triplets, that no difficulty arises from 
this circumstance, because in this case equality of properties with 
respect to three directions at right angles to each other will suffice 
for our purpose. Arrangements possessing this limited kind of iso- 
tropy, can easily be imagined for different numbers of corpuscles, pro- 
vided there be at least four of them. The electrons may be placed 
at the angles of one of the regular polyhedra, or of a certain number 
of such polyhedra whose centres coincide with that of the positive 
sphere, and whose relative position presents a sufficient regularity. 

Our final remark relates to the radiation emitted by the atom. 
When we examined the radiation from a single electron we found 
that it is determined by the acceleration. One can infer from this 
that the radiation produced at distant points by an atom which 
contains a number of equal vibrating electrons, and whose dimensions 
are very small in comparison with the wave-length, is equal to that 
which would take place if there were but one electron, moving with 
an acceleration that is found by compounding all the individual 
accelerations. In some cases, especially likely to occur in systems 
presenting a geometrical configuration of high regularity, this resultant 
acceleration is zero, so that there is no perceptible radiation at all, or 
at least only a very small residual one, due to the fact that the 
different electrons are not at exactly the same distance from the 
outer point considered, and that therefore we have to compound the 
accelerations, such as they are, not at one and the same instant, but 
at slightly different times. Vibrations presenting the peculiarity in 
question may properly be designated as ineffective ones. 

100. We shall now occupy ourselves with a special case, the 
simplest imaginable, namely that of four equal electrons A, B, C, D, 
which, of course, are in equilibrium at the corners of a regular 
tetrahedron whose centre coincides with the centre of the positive 
sphere.-*-) 

The fundamental modes of motion of this system can easily be 
determined.^) In order to obtain simple formulae for the frequencies, 

1) The Ze em an- effect in a system of this kind has already been examined 
by J. J. Thomson, who, ho-wever, supposed the positive sphere to have a uni- 
form volume-density. 

2) Note 48. 



VIBRATIONS OF A SYSTEM OF FOUR ELECTRONS. 121 

I shall imagine a spherical surface to pass through A, B, C, D; 
I shall denote by q the value which the density of the positive 
charge presents at this surface, and by p,, the mean density in its interior. 
I shall further introduce a certain coefficient m, which, in those 
cases in which there is a Zeeman-effect, can be regarded as a 
measure of it. We shall be concerned only with triplets, and the 
meaning of ra is, that the actual separation of the components is 
found, if the separation required by the elementary theory, for the 

same value of - , is multiplied by co. 

In the first fundamental mode, the four electrons perform equal 
vibrations along the lines OA, OB, OC, OD, in such a way that, 
at every instant, they are at equal distances from the centre 0. The 
frequency of this motion, which is inefficient, and not affected by a 
magnetic field, is determined by 



n' = — 

m ■ 



a formula which gives a real value for n, because q is positive and 
e negative. 

Other modes of motion are best described by choosing as axes 
of coordinates the lines joining the middle points of opposite edges 
of the tetrahedron, and by fixing our attention on two such edges, 
for example on those which are perpendicular to OX Let these 
edges be AB and CD, x being positive for the first, and negative 
for the second. 

The corpuscles can vibrate in such a manner that, at every 
instant, the displacement of any one of them from its position of 
equilibrium can be considered as made up of a component ji parallel 
to OX and a transverse component, which for A and B is along 
AB, and for G and D along GJ). CalKng the component j) positive 
or negative according to its direction, which may be that of OX or 
the reverse, and giving to the transverse displacement the positive 
sign if it is away from OX, and the negative sign if it is directed 
towards this line, we have for all the electrons 

p = a cos nt, 

for the transverse displacement of A and B 

and for that of G and D 

-g = - sp, 

the constant s being determined by the equation 

s = V 1/2 + 1/TT2VS (181) 



122 ni. THEORY OF THE ZEEMAN-EFFECT. 

where 

2? — po 



8(e-eo) 



(182) 



1 

CO = — 


[l - «-^ " 


4 

1 

CO = -7- 


L 1/2(1 + 20;^) J 


4 


L 1/2(1 + 2 ^^)J 



The double sign in (181) shows that there are tivo modes of the 
kind considered. These have unequal frequencies, for which I find 
the formula 

''" = - ifm { 6? - Po + 4 (P - Q,)V2{y+2v^)], (183) 

and are both effectiYC for radiation, on account of the accelerations 
of the electrons in the direction OX. The system will therefore 
produce two lines L^ and ig i^ ^^ spectrum. 

Now, it is immediately seen that, in addition to these two modes 
of vibrating, which are related, as we may say, to the direction OX, 
there are similar ones related in the same way to 01^ and OZ, so 
that ij and L^ are triple lines, which can be split into three com- 
ponents by a magnetic field. For Lj, the separation between the 
outer components and the middle one is determined by 

(184) 

and for L^ by 

(185) 

Moreover, it can be shown that the state of polarization of the hght 
producing the components of these triplets is the same which we 
have deduced from the elementary theory, the radiation along the 
lines of force again consisting of two circularly polarized beams of 
different frequencies, the one right-handed and the other left-handed. 
The modes of motion to which I have next to call your attention 
may be described as a twisting of the system around one of the 
axes OX, OY, OZ. The first of these modes is characterized by 
small rotations of the lines AB and CD around the axis OX, the 
direction of the rotation changing periodically for each line and being 
at every instant opposite for the two lines. Since a twisting of this 
kind around OZ can be decomposed into a twisting around OX and 
one around 01", these motions constitute only two fundamental 
modes. They are ineffective, and their frequency, which is given by 
the formula 

is not altered by a magnetic field. 

We have now found nine fundamental modes of motion in all. 
The remaining ones are rotations around one of the axes OX, 



VIBRATIONS OF A SYSTEM OF FOUR ELECTRONS. 123 

OY, OZ; these are not controRed by the internal forces we have 
assumed, and cannot be caUed vibrations about the position of 
equilibrium. 

101. It is worthy of notice, that (186) always gives a real value 
for n, and that the two frequencies determined by (183) are real too, 
provided the value of p be greater than ~q^. When this condition 
is fulfilled, the original state of the system is one of stable equilibrium. 

If we adopt J. J. Thomson's hypothesis of a uniformly charged 
sphere, we have Q = g^. In this case we can write instead of flSS), 
(184) and (185) i) ^ ^' 

2 m 3 TO ' 

1 
05 = — Y or 1 . 

102. Other cases in which a certain number of electrons have 
a regular geometrical arrangement within the positive sphere, can be 
treated in a similar way, though for a larger number of particles 
the calculations become rather laborious. So far as I can see, the 
liue of thought which we are now following promises no chance of 
finding the explanation of a quartet or a quintet, so that, after all, 
the progress we have made is not very important. The main interest 
of the preceding theory lies in the fact, that it shows the possibility 
of the explanation of magnetic triplets in which the separation of 
the components is different from that of the triplets of the elementary 
theory, as is shown by the value of a differing from 1. According 
to our formulae, o can even have a negative value. In the above 
example this means that, in the radiation along the lines of force, 
the circular polarization of the outer components of the doublet can 
be the inverse of what it would be according to the elementary 
theory.^) 

It is remarkable that negative electrons may in this way produce 
a Zeeman-effect which the elementary theory would ascribe to the 
existence of movable positive particles. 

103. Shortly after Zeeman's discovery some physicists observed 
that, just like the magnetic rotation of the plane of polarization, the 
new phenomenon makes one think of some rotation around the lines 
of force, going on in the magnetic field. There is certainly much to 
be said in favour of this view. Only, if one means the hidden ro- 



1) Note 49. 2) Note 60. 



124 ni. THEORY OF THE ZEEMAN-BFFECT. 

tations which some theories suppose to exist in the ether occupying 
a magnetic field (and to which those theories must ascribe every 
action of the field) a development of the idea lies outside the scope 
of the theory of electrons as I am now expounding it, because, in 
this theory, we take as our basis, without further discussion, the 
properties of the ether which are expressed in our fundamental 
equations. There is, however, a rotation of a different kind to which 
perhaps we may have recourse in our attempts to explain Zeeman s 
phenomenon. 

Let us consider the interval of time during which a magnetic 
field is set up in a certain part of the ether. While the magnetic 
force H is changing, there are electric forces d, whose distribution 
and magnitude are determined by our fundamental equations (2) and (5). 
These are the forces which cause the induction current produced in 
a metallic wire, and they may be said to be identical, though 
presented in a modem form, with the forces by which W. Weber 
explained the phenomena of diamagnetism, an explanation that can 
readily be reproduced in the language of the theory of electrons. 
I shall now consider the rotation they impart to a system of negative 
electrons such as we have been examining in the preceding paragraphs. 
In doing so, I shall suppose the positively charged sphere to have 
so large a mass that it may be regarded as immovable, and I shall 
apply to the system of negative electrons the laws that hold for a 
rigid body; this will lead to no appreciable error, if the time during 
which the magnetic field is started, is very long in comparison with 
the periods of the vibrations of the electrons. 

104:. I shall again confine myself to arrangements of the electrons 
that are isotropic with respect to three directions at right angles to 
each other. Then, if the axes of coordinates are drawn through the 
centre in any directions we like, and if the sums are extended to 
all the negative electrons of the system, we shall have 

^x = ^y = ^s = 0. 

Also, the moment of inertia will be the same with respect to any 
axis through 0. We may write for it 

Q = 2mK, 
if 

and we have I (187) 

^^2/ = ^y^ = ^soc = 0. 
The force acting on one of the electrons is given by 

ed,, ed„, ed,, 



dx dyj 



ROTATION OF A PARTICLE IN A MAGNETIC FIELD. 125 

and we find therefore for the components of the resultant couple 
with reference to the point 

e2{y^,-^^y), e^(^d, - a;tl,), e^(:rd^ - j/dj. (188) 

By d we shall understand the electric force due to causes out- 
side the system. On account of the small dimensions of the latter, 
this force will be nearly constant throughout its extent, so that, 
denoting by d^ the electric force at the centre, we may write 

Substituting these values in the expressions (188), and bearing in 
mind the equations (187), one finds 

or, in virtue of the fundamental equation (5), 

--KK, --Kin,,, --Kin^. 

c ^' c y' c ' 

In order to find the components of the angular acceleration, we must 
divide these expressions by § = 2mK. The result is 

~ iinc ^' ~ 'imc v^ ~ 2mc "' 

from which it at once appears that, after the establishment of a 
field H, a system that was initially at rest, has acquired a velocity 
of rotation 

k = - ~ H. (189) 

The axis of rotation has the direction of the magnetic field, and, if 
e is negative, the direction of the rotation corresponds to that of the 
field. It is interesting that the velocity of the rotation is independent 
of the particular arrangement of the electrons, and that its frequency, 
i. e. the number of revolutions in a time 2%, is equal to the change 
of frequency we have calculated in the elementary theory of the 
Zeeman- effect. 

The same rotation would be produced if, after the setting up of 
the field, the system were, by a motion of translation, carried into 
it from an outside point. Once started, the rotation will go on for 



126 ni. THEORY OF THE ZEEMAN-EFFEGT. 

ever, as long as the field is kept constant, unless its velocity be 
slowly diminished by the radiation to which it gives rise.^) 

105. We shall now turn our attention to the small vibrations 
that can take place in the system while it rotates. For this purpose, 
we shaU introduce axes of coordinates having a fixed position in the 
system, and distinguish between the motion with respect to these 
axes, the relative motion, and the motion with respect to axes fixed 
in space, which we may call the absolute one. 

Let, for any one of the electrons, V be the absolute velocity, 
q the absolute acceleration, q^ the part of it that is due to the 
internal forces of the system, and (\^ the part due to the magnetic 
field. Then, we shall have for the acceleration q' of the relative 
motion, if we neglect terms depending on the square of the angular 
velocity k, and therefore on the square of the magnetic force H, 

q' = q-2[kv] = qi + q, -2[kv], 

i. e. on account of (189), 

q' = q. + q. + ,-;f^[H v]. 
Since 

we find^) 



me ' 



q =qi- 

This shows that the relative motion is determined solely by the 
internal forces of the system; it is identical with the motion that 
could take place in a system without rotation and free from the 
influence of a magnetic field. I shall express this by saying that in 
the system rotating with the velocity which we calculated, there is 
no internal Zeeman-effect, the word „internal" being introduced, 
because, as we shall presently show, there remains a Zeeman-effect 
in the external radiation. This effect is brought about by the same 
cause that has made the internal effect disappear, namely by the 
rotation of the particles. 

106. We have already observed (§ 99) that a particle which 
contains a certain number of equal vibrating electrons, and whose- 
size is very small compared with the wave-length, will radiate in the 
same way as a single electron of the same kind, moving with the 
accelerations ^x, ^y , 2'^, the sums extending to all the separate 
electrons, and x, y, s being their coordinates with respect to axes 

1) Note .51. 2) Note .52. 



ZEEMAN-EFFBCT DUB TO THE ROTATION OP A PARTICLE. 127 

fixed in space. The accelerations will have these values if the co- 
ordinates of the equivalent electron, as it may properly be called, 
are given at every instant by ^x, ^y, "Ss. 

In order to apply this theorem to the problem before us, I shall 
again choose the centre of the positive sphere as origin of coordinates, 
drawing the axis of s in the direction of the external magnetic 
force H. Let OX and OF be fixed in space, and let OX', OY' be 
axes rotating with the system; then, if h is the positive or negative 
velocity of rotation around OZ, we may put 

X = x cos kt — tf sin Id, \ 

, \ (190) 

y = x' sin ht + y cos lit, J 

since we may take Ttt for the angle between OX and OX'. Now, 
if Xfl', 2/g', 5q' are the coordinates of one of the negative electrons in 
its position of equilibrium, and a cos {nt-\-f), ^ cob (nt -\- g), 
y cos (nt -\- h) the displacements from that position, due to the 
internal vibrations, and referred to the moving axes, we shall have 
for this particle 

x'=Xq' + K cos (nt + f), y' = y^' + p cos {nt + g). (191) 

Whereas the constants a, /3, f, g (and y, h) have different values for the 
several electrons, the frequency n will have for all these corpuscles 
a common value, equal to the frequency of the radiation in the 
absence of a magnetic field. 

Introducing the values (191) into the expressions (190) and 
taking the sum for all the corpuscles, we shall find the coordinates 
X, y of the equivalent electron. Since ^x^ = ^y^ =0, the result 
may be put in the form 

X = Xj + Xj, y = yi + 7i, 
where 

:s^ = A cos {(n + 'k)t + (p] , y^= A sm [{n -\- li)t -{- cp] , 

X2 = JJcos [{n-l)t+il>\, y^ = —Bsva.[{n-'k)t-\-il!), 

A, B, q) and j/; being constants. These formulae show that, leaving aside 
the vibration in the direction of OZ, which is entirely unaffected by 
the field and the rotation, we can decompose the motion of the 
equivalent electron into two circular motions in opposite directions, 
performed with the frequencies n + h and n — I: Therefore, since 
in virtue of (189) Ic is given by the equation 

2mc' ' 
the Zeeman-effect in the radiation issuing from the rotating particle 
exactly corresponds to that which we formerly derived from the 
elementary theory for a particle without rotation. 



128 III. THEORY OF THE ZEEMAN-EFFECT. 

107. There are one or two points in this last form of the 
theory that are particularly to be noticed. 

In the first place, we can suppose the system of electrons within 
the positive sphere to be capable of vibrating in different modes, 
thereby producing a series of spectral lines. In consequence of the 
rotation set up by the field, all these lines wUl be changed into 
equal triplets, so that we have now found a case, in which aU the 
lines of a series are divided, as they reaUy are, in the same way. 
I may add that, according to the view of the phenomenon we are 
now discussing, the Zeeman-effect is due to a combination of the 
internal vibrations whose frequency is n, with the rotation of the 
frequency k. 

This calls forth a more general remark. It is well known that 
in acoustic phenomena two tones with the frequencies n^ and Wg 
are often accompanied by the so called combination-tones whose 
frequencies are n^ + n^ and n^ — n^ respectively. Something of the 
same kind occurs in other' cases in which a motion or any other 
phenomenon shows two different kinds of periodicity at the same 
time; indeed, on account of these, terms such as cos n.^i and cos n^t 
wiU occur in the mathematical expressions, and as soon as the product 
of two quantities having the two periods shows itself in the formulae, 
the simple trigonometric formula 

cos %^ COS n^t = -— cos (Wj + Mg)^ + ^ cos (n^ — n^)t 

leads us to recognize two new frequencies n^ + '*2 ^^^ ''h ^ ^2- 
Indeed, it is precisely in this way that, in the preceding paragraph, 
the frequencies w + A; and n — li have made their appearance. 

Many years ago, V. A.. Julius observed that certain regularities 
in the spectra of elements may be understood, if we suppose the 
lines to be caused by combination -tones, the word being taken in 
the wide sense we can give it on the ground of what has just been 
said. If, for example, there are two fundamental modes of vibration 
with the frequencies n^ and n^ or, as we may say more concisely, 
two „tones" n^ and n^, and if each of these combines with a series 
of tones, so that secondary tones with frequencies equal to the diffe- 
rences between those of their primaries are produced, we shall obtain 
a series of pairs, in which the components of each pair are at the 
distance n^ — n^ from each other. 

In connection with this, it should also be noticed that, in 
Rydberg's formulae, every frequency is presented as the difference 
between two fundamental ones. 

Of course it would be premature to attach much value to 
speculations of this kind. Yet, in view of the fact that all lines of 



SECONDARY VIBRATIONS. 129 

a series undergo the same magnetic splitting, one can hardly help 
thinking that all the fundamental modes of motion helonging to the 
series are somehow combined with one or more periodic phenomena 
going on in the magnetic field, as, in the example we have worked 
out, they were combined with the rotation of the particles. 

I may add that the form of Ryd berg's equations, in which each 
frequency is represented as the difference of two terms, naturally 
suggests the idea that under the influence of a magnetic field one 
or both of these terms have their value changed, or rather, are replaced 
by a number of slightly different terms, to each of which corresponds 
a magnetic component. It is clear that if, for all lines of a series, 

the part of ^ which they have in common, for instance the part 



in the second subordinate series I (§ 82), is altered in the same way, 
the other part — . .; remaining unchanged, the equality of the 
Zeeman-effect for all the members of the series wiU be accounted for. 

108. In the second place, it is important to remark that, for 
the entire prevention of an internal Zeeman-effect, the rotation of 
a particle as a whole must have exactly the velocity we have found 
for it in § 104. 

For other values of k, such as might occur if the rotating par- 
ticle had a moment of inertia diS'erent from that which we formerly 
took into account, q' would come out different from q^, so that the 
relative motion of the electrons with respect to the rotating axes 
would still be affected by the magnetic force. In such a case, in 
order to find the Zeeman-effect as it becomes manifest in the 
radiation, we should have to combine the internal motions with the 
rotation, after the manner shown in § 106; the result would obviously 
be a decomposition of the original spectral lines into more than 
three components. 

This seems rather promising at first sight, but it must be 
admitted that one can hardly assign a reason for the existence of a 
moment of inertia, different from the value used in § 104, and that 
it would be very difficult to reconcile the results with Runge's law 
for the multiple divisions of the lines. 

109. The preceding theory of rotating radiating particles is 
open to some objections. Besides the two cases mentioned in § 104, 
a third must perhaps be considered as possible. In a Geissler 
tube or a flame combinations and decompositions of minute particles 
are no doubt continually going on; a radiating atom cannot there- 
fore be supposed to have been in a free state ever since the magnetic 
field was set up. Now, in atoms combined with other particles, the 

Loreutz, Theory of electrons. 9 



130 m. THEORY OF THE ZEEMAN-BFFECT. 

mobility of the electrons might perhaps be so much diminished, that 
the production of the field cannot make them rotate; since there is 
no reason why they should begin to do so the moment the atoms 
are set free, we can imagine in this way the existence in the mag- 
netic field of free atoms without a rotation. 

Another difficulty, which one also encounters in some questions 
belonging to the theory of magnetism, arises from the fact that a 
rotating particle whose charge is not quite uniformly distributed, 
must necessarily, in the course of time, lose its energy by the 
radiation that is due to the rotation itself. It is probable that 
the time required for an appreciable diminution of the rotation 
would be Tery long An exact determination of it would, however, 
require rather complicated calculations. 

110. After aU, you see by these considerations that we are 
rather at a loss as to the explanation of the complicated forms of 
the Zeem an -effect. In this state of things, it is interesting that 
some conclusions concerning the polarization of the radiation can be 
drawn from general principles, independently of any particular theory. 
For this purpose we shall avail ourselves of the consideration of 
what we may term the reflected image of an electromagnetic system. 

Let iS be a system composed of moving electrons and material 
particles, the motion of the former being accompanied by an electro- 
magnetic field in the intervening ether. Then, a second system S', 
which may be called the image of S with respect to a plane F, may 
be defined as follows. To each particle or electron, and even to 
each charged element of volume in S, corresponds an equal particle, 
electron or element of volume in S', moving in such a way that the 
positions of the two are at every instant symmetrical with respect 
to the plane V; further, if P and P' are corresponding points, the 
vector representing the dielectric displacement at P' is the image of 
the corresponding one at P, whereas the magnetic forces in S and 
S' are represented by vectors, one of which is the inverted image of 
the other. On certain assumptions concerning the forces between the 
electrons and other particles, which seem general enough not to 
exclude any real case, the system S' can be shown to be a possible 
one, as soon as S has an objective existence. 

We shall apply this to the ordinary experiment for the 
exhibition of the Zeeman-effect, fixing our attention on the rays that 
are emitted along the lines of force, and placing the plane V parallel 
to these lines. There are many positions of the plane fulfilling the 
latter condition, but it is clear that, whichever of them we choose, 
the image of the electromagnet will always have the same properties. 
The same may be said, so far as the properties are accessible to our 



POLARIZATION OF THE RADIATION. 131 

observations, of the source of light itself; therefore, the radiation 
too must be exactly the same in all systems that can be got by 
taking the image of the experiment with respect to planes that are 
parallel to the lines of force. From this we can immediately infer 
that the light radiated along the lines of force can never show a 
trace of rectilinear or elliptic polarization; it must either be un- 
polarized, or have a circular polarization, partial or complete. This 
conclusion also holds for the part of the radiation that is charac- 
terized by a definite frequency, and is therefore found at a definite 
point of the spectrum. 

By a similar mode of reasoning we can predict that, in the 
emission at right angles to the lines of force, there can never be 
any other polarization, either partial or complete, but a rectilinear 
one with the plane of polarization parallel or perpendicular to the 
lines of force. 

Finally, since the image of a magnetic field with respect to the 
plane of which we have spoken, is a field of the opposite direction, 
the state of radiation must be changed into its image by an inversion 
of the magnetic force. At every point of the spectrum the direction 
of the circular polarization will be inverted at the same time. 



CHAPTER IV. 

PROPAGATION OF LIGHT IN A BODY COMPOSED OF 

MOLECULES. 

THEORY OF THE INVERSE ZEEMAN-EFFECT. 

111. In the preceding discussion we had in view the influence 
of a magnetic field on the light emitted by a source of light. There 
is a corresponding influence on the absorption, as was already shown 
by one of Zeeman's first experiments. He found that the dark Hues 
which appear in the spectrum of a beam of white light, passed 
through a sodium flame, are changed in exactly the same way as 
the emission lines of the luminous vapour, when the flame is exposed 
to an external magnetic field. We can easily understand this inverse 
phenomenon if we bear in mind the intimate connexion between the 
emission and the absorption of light. According to the well known 
law of resonance, a body whose particles can execute free vibrations 
of certain definite periods, must be able to absorb light of the same 
periods which it receives from without. Therefore, if in a sodium 
fiame under the influence of a magnetic field there are three periods 
of free vibrations instead of one, we may expect that the flame can 
produce in a continuous spectrum three absorption lines corresponding 
to these periods, and in general, if we want to know what kinds of 
light are emitted by a body imder certain circumstances, we have 
only to examine the absorption in a beam of light sent across it. 

A highly interesting theory based on this idea has been deve- 
loped by Voigt.^) It has the advantage of being applicable to bodies 
whose density is so great that there is a certain mutual action 
between neighboring molecules, a case in which it is rather difficult 
directly to consider the emission of light. 

Voigt's theory was not originally expounded in the language 
of the theory of electrons, his first method belonging to those which 



1) W. Voigt, Arm. Phys. Chem. 67 (1899), p. 346; 68 (1899), p. 362, 604; 
69 (1899), p. 290; Ann. Phys. 1 (1900), p. 376, 389; 6 (1901), p. 784; see also 
his book: Magneto- und Elektrooptik, Leipzig, 1908. 



MEAN VALUES. 133 

I have formerly alluded to, in whicli one tries to describe the observed 
phenomena by judiciously chosen differential equations, without 
troubling oneself about the mechanism underlying them. However, 
in order not to stray from the main subject of these lectures, I shall 
establish Voigt's equations, or rather a set of formulae equivalent 
to them, by applying the principles of the theory of electrons to the 
propagation of light in a ponderable body considered as a system of 
molecules. 

These formulae are also interesting because by means of them 
we can treat a number of other questions, relating to the velocity 
of propagation and the absorption of light of different frequencies. 
It will be well to begin with some of these, deferring for some time 
the consideration of the action exerted by a magnetic field. 

112. Let us imagine a body composed of innumerable molecules 
or atoms, of „particles" as I shall term them, each particle con- 
taining a certain number of electrons, all or some of which are set 
vibrating by an incident beam of light. Between the electrons and 
in their interior there will be a certain electromagnetic field, which 
we could determine by means of our fundamental equations, if the 
distribution and the motion of the charges were known; having 
calculated the field, we should also be able to find its action on the 
movable electrons, and to form the equations of motion for each of 
them. This method, in which the motion of the individual electrons 
and the field in their immediate neighborhood and even within 
them, would be the object of our investigation, is however wholly 
impracticable, when, as in gaseous bodies and liquids, the distribution 
of the particles is highly irregular. We cannot hope to follow 
in its course each electron, nor to determine in all its particulars 
the field in the intermolecular spaces. We must therefore have 
recourse to an other method. Fortunately, there is a simple way of 
treating the problem, which is sufficient for the discussion of what 
can really be observed, and is indeed suggested by the very nature 
of the phenomena. 

It is not the motion of a single electron, nor the field produced 
by it, that can make itself felt in our experiments, in which we are 
always concerned with immense numbers of particles ; only the resultant 
effects produced by them are perceptible to our senses. It is to be 
expected that the irregularities of which I have spoken, will disappear 
from the total effect, and that we shall be able to account for it, 
if, from the outset, we fix our attention, not on all these irregularities, 
but only on certain mean values. I shall now, proceed to define these. 

113. Let P be a point in the body S a sphere described 
around it as centre, and qp one of the scalar or vector quantities 



134 IV. PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES. 

occurring in our fundamental equations. Then the mean value of g) 
at the point F, which we shall denote by ^, is given by the equation 



f = -qJ fiiS, 



in which S means the volume of the sphere, and the integration is 
to be extended to this volume. The elements dS are to be taken 
infinitely smaU in the mathematical sense of the words, so that even 
an electron is divided into many elements. As to the sphere S, it 
must be chosen neither too small nor too large. Since our purpose 
is to get rid of the irregularities in the distribution of q), the sphere 
must contain a very large number of particles. On the other hand, 
we must be careful not to obliterate the changes from point to point 
that can really be observed. The radius of the sphere must there- 
fore be so small that the state of the body, so far as it is accessible 
to our means of observation, may be considered as uniform throughout 
the sphere. In the problems we shall have to deal with, this means 
that the radius must be small compared with the wave-length. For- 
tunately, the molecular distances are so much smaller than the length 
of even the shortest light-waves, that both conditions can be satisfied 
at the same time. 

114. The mean value ^, taken for a point P, is in general a 
function of the coordinates of this point, and if (p itself depends 
upon the time, ip will do so too. We can easily infer from our 
definition the relations 

dcp dcp df dq> 

dx~dx> • ■' Jt^Jt' 

by which the transition from our fundamental equations to the 
corresponding formulae for the mean values is made very easy. Of 
course, the mean values of the quantities on the right -and on the left- 
hand side of an equation must be equal to each other, so that all 
we have got to do, is to replace d, h, etc. by their mean values. 
The resultant fonnulae, viz. 

rot h = — (d + pv) 
and 

rot d = h, 

may be considered as the general electromagnetic equations for the 
ponderable body; they are comparable with those of which we spoke 
in § 4. In order to bring out the similarity, I shall put 



GENERAL EQUATIONS FOR MEAN VALUES . 135 

and 

h = H. 

It only remains to examine the term pv. According to our definition 
of mean values, we have for the components of this vector, if x, 
y, s are the coordinates of an element of the moving charges at the 
time t, 

or, if we suppose the surface of the sphere not to intersect any 
electrons. 

We have formerly seen that I QxdS, j Qi/dS, j QZdS are the com- 
ponents of the electric moment of the part of the hody to which 

the integration is extended. Hence, ^ / QxdS and the two corre- 
sponding expressions with y and z are the components of the electric 
moment of the body per unit of volume.^) We shall represent this 
moment, or, as it may also be termed, the electric polarization of the 
body, by P Thus 

PV = P, 

and 

d + 9V = E + P. 

Simplifying stOl further by putting 

E + P = D, (192) 

we are led to the equations 

rotH = -D, (193) 



c 



rotE = -|H, (194) 

which have exactly the form of those of which we spoke in § 4. 
If we like, we may now call E and D the electric force and the 
dielectric displacement, D the displacement current. This exactly 
agrees with common usage; only, in our definition of these vectors, 
one clearly sees the traces of our fundamental assumption that the 
system is made up of ether and of particles with their electrons. 
Thus, E is the mean force acting on a charge that is at rest. The 
total dielectric displacement D consists of two parts, the one E having 
its seat in the ether, and the other P in the particles. Corresponding 



1) Note 63. 



136 IV. PROPAGATION OP LIGHT IN A SYSTEM OF MOLECULES. 

to these, we distinguish two parts in the current D; the first, E is 
the mean of the displacement current in the ether, and the second 
the mean of the conyection current qV. 

115. To complete our system of equations we must now 
examine the relation between D and E, or rather that between P 
and E. This is found by considering the way in which the electric 
moment in a particle is produced or changed. 

Let us suppose that each particle contains a single movable 
electron with charge e and mass m, and let us denote by |, tj, ^ the 
distances over which it is displaced from its position of equilibrium 
in the directions of the axes of coordinates. The components of the 
electric moment of a single particle are 

and, writing N for the number of particles per unit volume, we have 

P, = Ne^, ?,j = Neri, ?.^ = Nei, (195) 

if the particles have a regular geometrical arrangement. If, on the 
contrary, they are irregularly distributed, so that the values of the 
displacements |, y], t, change abruptly from one particle to the next, 
we may use the same equations, provided we understand by |, t], t, 
mean values taken for all the particles situated in a space that is 
infinitely small in a physical sense. A similar remark applies to 
other quantities occurring in the equations we are going to establish 
for the motion of the electrons. 

116. The values of |, rj, g, and consequently those of P^, P^,, P^, 
depend on the forces acting on the movable electrons. These are of 
four different kinds. 

In the first place we shall conceive a certain elastic force by 
which an electron is pulled back towards its position of equilibrium 
after having been displaced from it. We shall suppose this force to 
be directed towards that position, and to be proportional to the 
displacement. Denoting by /' a certain positive constant which 
depends on the structure and the properties of the particle, we write 
for the components of this elastic force 

-n, -h, -ft (196) 

The second force is a resistance against the motion of the 
electron. We must introduce some action of this kind, because 
without it it would be impossible to account for the absorption 
which it is one of our principal objects to examine. Following the 
example given by Helmholtz in his theory of anomalous dispersion, 



FORCES ACTING ON AN ELECTRON. 137 

with which the present investigation has many points in common, 
I shall take the resistance proportional to the velocity of the electron, 
and opposite to it. Thus, if </ is a new positive constant, the com- 
ponents of the second force are 

-«% -of,' -«§■ (li") 

We shall later on return to this question. 

117. We have next to consider the force acting on the electron 
on account of the electromagnetic field in the ether. At first sight 
it may be thought that this action is to be represented by eE. On 
closer examination one finds however that a term of the form aeP 
is to be added, in which a is a constant whose value is little different 
from •^. I shall not enter upon the somewhat lengthy calculations 
that are required for the determination of this additional force. In 
order to explain why it is introduced, I have only to remind you 
of the well known reasoning by which Kelvin long ago came to 
distinguish between the magnetic force and the magnetic induction. 
He defined these as forces exerted on a pole of unit strength, placed 
in differently shaped infinitely small cavities surrounding the point 
considered. The magnetically polarized parts of the body outside the 
cavity turn their poles more or less towards it, and thus produce on 
its waUs a certain distribution of magnetism, whose action on an in- 
side pole is found to depend on the form of the cavity. 

In the problem before us we can proceed in an exactly similar 
manner. The general equations (33) — (36) show that the electro- 
magnetic field is composed of parts that are due to the individual 
particles of the system, so that, if some of these were removed, the 
motion of the electrons in the remaining ones being left unchanged, 
a part of the field would be taken away. We must further take into 
account that each component of d or h belonging to the field that 
is produced by a certain number of particles, is obtained by an 
addition of the corresponding quantities for the fields due to each of 
the particles taken separately. The sum may be replaced by an 
integral in those cases in which the discontinuity of the molecular 
structure does not make itself felt. If we want to know the field 
produced at a point J. by a part of the body whose shortest distance 
from A is very great compared with the mutual distance of adjacent 
particles, we may replace the real state of things by one in which 
the polarized matter is homogeneously distributed. 

AU this can also be said of the magnetized particles one has to 
consider in Kelvin's theory, though the cases are diS'erent, because 
the formulae (§ 42) for the field produced by a variable electric 



138 IV. PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES, 

moment are less simple than those which determine the action of a 
constant molecular magnet. The formulae however much resemble 
each other if the point for which the field of a particle is to be 
determiaed, lies at a distance from it that is small compared with 
the wave-length. In this case the field can be approximately con- 
sidered as an electrostatic one, such as would exist if the electric 
moment did not change in the course of time. 

Around the particle A for which we wish to determine the 
action exerted on the electron it contains, we lay a closed surface (?, 
whose dimensions are infinitely small in a physical sense, and we 
conceive, for a moment, all other particles lying within this surface 
to be removed. The state of things is then exactly analogous to the 
case of a magnet in which a cavity has been formed. There will be 
a distribution of electricity on the surface, due to the polarization of 
the outside portion of the body, and the force E', exerted by this 
distribution on a unit charge at A must be added to the force E 
which appears in (194). 

Now, if the particles we have just removed are restored to their 
places, their electric moments will produce a third force E" in the 
particle A, and the total electric force to which the movable electron 
of A is exposed, will be 

E + E' + E" 

It is clear that the result cannot depend on the form of the cavity 6, 
which has only been imagined for the purpose of performing the 
calculations. These take the simplest form if ^ is a sphere. Then 
the calculation of the force E' leads to the result^) 

E' = |P 

The problem of determining the force E" is more difficult. I shall 
not dwell upon it here, and I shall only say that, for a system of 
particles having a regular cubical arrangement, one finds ^) 

E" = 0, 

a result that can be applied with a certain degree of approximation 
to isotropic bodies in general, such as glass, fluids and gases. It is 
not quite correct however for these, and ought to be replaced in 



general by 



E" = sP, 



where, for each body, s is a constant which it will be difficult 
exactly to determine. 



1) Note 54. 2) Note 66. 



EQUATIONS OF MOTION OF AN ELECTRON. 139 

Putting 

a = I + s, 

we find for the electric force acting on an electron 

E + aP. 

118. The last of the forces we are enumerating occurs in 
magneto-optical phenomena; it is due to the external magnetic field, 
which we shall denote by the symbol Q, in order to distinguish it 
from the periodically changing magnetic force H that is due to the 
electric Tibrations themselves'), and occurs in our equations (193) 
and (194). 

In all that follows we shall suppose the external field ^ to 
haye the direction of the axis of z. Then its action on the vibrating 
electron, which in general is represented by 

has the components 

e§ dri e§ (£| 



c dt' c dt' 



0, 



where Q is written instead of ^^. 

Taking together all that has been said about the several forces, 
we find for the equations of motion of the movable electron contained 
in a particle 



^dt^-^^^y^"'^y) '*' 9 dt c dt' 



(198) 



119. Another form of these equations will be found more con- 
venient for our purpose. 

In the first place, instead of the displacements of the movable 
electron, we shall introduce the components of the electric polari- 
zation P- Taking into account the relations (195), dividing the 
formulae (198) by e, and putting 

i^^ = -'^ 1.7^ = A ^. = 9, (199) 



1) Note 66. 



140 IV. PROPAGATION OP LIGHT IN A SYSTEM OF MOLECULES, 
one finds 

,a!P._F , p _,.p _„'SP^ ^^dPy 



m 






(200) 



These equations may be further transformed, if, in our investigation 
of the propagation of simple harmonic vibrations, we use the well 
known method in which the dependent variables in the system of 
equations are first represented by certain exponential expressions with 
imaginary exponents, the real parts of these expressions, to which 
one has ultimately to confine oneself, constituting a solution of the 
system. 

Let E be the basis of natural logarithms, and let aU dependent 
variables contain the time only in the factor 







SO that n is the frequency of the vibrations. 


Then, if we put 


a = f — a — m'n^, 


(201) 


/3 = ng, 


(202) 


^ cNe' 


(203) 



all real quantities, the formulae (200) take the form 

E. = (« + ^/3)P.-^>P,/ 

E, = (« + '/3)P, + ^yP., (204) 

E, = (« + i/3)P,. 

Since P = D — E, these equations may be said to express the relation 
between E and D which we have to add to the general formulae 
(193) and (194). 

120. Before coming to solutions of our system of equations, 
it will be well to go into some details concerning the cause by which 
the absorption is produced. We have provisionally admitted the 
existence of a resistance proportional to the velocity of an electron, 

which is represented by the terms —'J-f., —g-ri, ~9^ in (198) 

'' cit (it dt 

and by the terms i/3P^, jj3P,^, ijiP^ in (204). It must be observed, 

however, that in our fundamental equations there is no question of 

a resistance of this kind; as we have formerly seen, an electron can 

move for ever through the ether with undiminished velocity. In our 



RESISTANCE TO THE MOTION 141 

considerations we have come across only one force that may be 
termed a resistance, namely the force 

which is proportional to the rate of change of the acceleration. In 
the case of simple harmonic vibrations, its components can be re- 
presented in the form (197), with the following value of the coefficient 

^=S- (206) 

Some numerical data which I shall mention later on, show however 
that this force (205) is much too small to account for the absorption 
that is really observed in many cases. We must therefore look for 
some other explanation. 

It has occurred to me that this may be found in the assumption 
that the vibrations in the interior of a ponderable particle that are 
excited by incident waves of light, cannot go on undisturbed for ever. 
It is conceivable that the particles of a gaseous body are so pro- 
foundly shaken by their mutual impacts, that any regular vibration 
which has been set up in them, is transformed by the blow into the 
disorderly motion which we call heat. The rise in temperature 
produced in this way must be due to a part of the energy of the 
incident rays, so that there is a real absorption of light. It is also 
clear that the accumulation of vibratory energy in a particle, which 
otherwise, in the case of an exact agreement between the period of 
the vibrating electrons and that of the incident light, would never 
come to an end, will be kept within certain limits by this disturbing 
influence of the collisions, just as well as it could be by a resistance 
in the ordinary sense of the word. 

In working out this idea, one finds that the formulae we have 
established in what precedes may still be used, provided only we 
understand by g the quantity i) 

9-^-^, (207) 

in which t is the mean length of time during which the vibrations 
in a particle can go on undisturbed. Since we can use the same 
formulae as if there were a real resistance, it is also convenient to 
adhere to the use of the latter term, and to speak of the resistance 
originating in the collisions, this resistance becoming greater when 
the interval % is diminished. 

According to the above idea, the interval r ought, in gaseous 
bodies, to be equal to the mean length of time elapsing between two 

1) Note 67. 



142 IV. PROPAGATION OP LIGHT IN A SYSTEM OP MOLEGDLES. 

successive encounters of a molecule. Unfortunately, it is found that 
the value of t deduced from experimental data is smaller than the 
interval between two encounters. We must conclude from this that 
there are causes in the interior of a molecule by which the regularity 
of the vibrations is disturbed sooner than it would be by the molecular 
impacts. We cannot pretend therefore to have satisfactorily elucidated 
the phenomenon of absorption; its true cause remains yet to be 
discovered. 

121. Leaving aside for some time the effects produced by a 
magnetic field, we shall now examine the propagation of light in the 
case § = 0, y = 0. Let us first suppose that there is no resistance 
at all, so that (3 is likewise 0. Then the formulae (204) may be 
written 

E = «P, 
from which we deduce 

D = (i + 1)E. (208) 

Let the propagation take place in the direction of OZ, so that 
the components of E, D and H are represented by expressions con- 
taining the factor 

£'■"('-5*), (209) 

where g' is a constant. Then, since all differential coefficients with 
respect to x and y vanish, we have by (193) and (194) 

3« c 8t 
and 

8E^ ^ 1 gHy 

8z c 8t ' 

or 

gH^ = |D,, 2E, = |h,, 
whence 

Combining this with (208), we get 

cV=l+-^- (210) 

Supposing 1 + — to be positive, we find a real value for q. The 

real part of (209) is 

cosn(t — qz), 

from which it is seen that the velocity of propagation is 

_i_ 

~ 2 ' 



DISPERSION OF LIGHT. 143 

It can therefore be calculated by means of the equation (210), for 
which we may write 



if 



2 1 I ^ 



c 
u, = — 



is the index of refraction. 

It is to be noticed that our result agrees with Maxwell's well 
known law, according to which the refractive index of a body is 
equal to the square root of its dielectric constant. Indeed, the 
equation (208) shows that the ratio between the dielectric displace- 
ment D and the electric force E is given by 1 -| ; it is therefore 

this quantity which plays the part of the dielectric constant or the 
speciiic inductive capacity in Maxwell's equations. 

122. In one respect, however, the theory of electrons has enabled 
us to go further than Maxwell. You see from the equation (201) 
tbat, for a given system, a is not a constant, but changes with the 
frequency .n. Therefore, our formulae contain an explanation of the 
dispersion of light, i. e. of the fact that different kinds of light have 
not the same refractive index. 

This explanation is very much like that which was proposed by 
several physicists who developed the undulatory theory of light in 
its original form in which the ether was considered as an elastic 
body. Sellmeyer, Ketteler, Boussinesq and Helmholtz showed 
that the velocity of light must depend on the period of the vibra- 
tions, as soon as a body contains small particles which are set 
vibrating by the forces in an incident beam of light, and which are 
subject to intramolecular forces of such a kind that they can perform 
free vibrations of a certain definite period. The amplitude of the 
forced vibrations of these particles, which is one of the quantities 
determining the velocity of propagation, wiU largely depend on the 
relative lengths of their ovm period of vibration and the period of 
the light falling on them. The theory of the propagation of light 
in a system of molecules which has been here set forth, is based on 
the same principles as those older explanations of dispersion, the 
only difference being that we have constantly expressed ourselves in 
the terms of the electromagnetic theory, and that the small particles 
imagined by Sellmeyer have now become our electrons. 

If we conceive a single particle to be detached from the body, 
so that it is free from all external influence, and if we leave out of 
account the resistance which we have represented by means of the 
coefficient g, the equations of motion (198) simplify to 



144 IV. PROPAGATION OF LIGHT IN A SYSTEM OP MOLECULES. 

from which it appears that the electron can perform free vibrations 
with a frequency w^ determined by 

Introducing this quantity, and using (199), we may write instead of 
(201), if we put a = ^, 

a = min,^-n^) - | = ^(V"'^^) - | " (^H) 

The index of refraction is therefore determined by 

^^-1 = ^- (212) 

TO , . „. 1 

The value of fi derived from this formula is greater than 1, if 
the frequency n is so far below that of the free vibrations w, that 
the denominator is positive; if this condition is satisfied, we can 
further conclude that (i increases with the frequency. This agrees 
with the dispersion as it is observed in transparent bodies, at least 
if we suppose that in these the frequency n^ corresponds to rays in 
the ultra-violet part of the spectrum. 

123. As a further application of our results we can take the 
old problem of the connexion between the index of refraction fi of 
a transparent body, and its density q. As is well known, Laplace 
inferred from theoretical considerations, based on the form the 
undulatory theory had in his time, that, when the density of a body 
is changed, the expression 

^ (213) 

should remain constant. In most cases the observed changes of the 
refractivity do not at all conform to this law, and it has been found 
that a better agreement is obtained if Laplace's rule is replaced by 
the empirical formula 

^^^^ = const. (214) 

The electromagnetic theory of light leads to a new form of the re- 
lation. Indeed, by a slight modification, (212) becomes 

TO . 2 „, /i'-|-2 



For a given body and a given value of n, the expression 

(1^ + 2 



RELATION BETWEEN REFRACTIVITT AND DENSITY. 145 

must therefore be proportional to the number of molecules per unit 
volume, and consequently to the density. 

This result had been found by Lorenz^) of Copenhagen some 
time before I deduced it from the electromagnetic theory of light, 
which is certainly a curious case of coincidence. 

124. In a certain sense the formula may be said to be much 

older. Putting in (201) w = and, as before, a = y, we find for the 

case of extremely slow vibrations, or of a constant field 

_^, _£__/; J^ 

"~' 3 "~ JVe^ 3 ■ 

The corresponding value of the ratio 1 -| between D and E is 

1 



B=l + 



_L 1 

Ne^ 3 



This, therefore, is the value of the dielectric constant for our system 
of molecules, a result which we could also have obtained by a direct 
calculation. 

Now, the last formula shows that, when N is changed, the value of 

i — X 

remains constant. Hence, the relation between the dielectric constant 
and the density q is expressed by 

/ I o, = const., 

(s + 2)e ' 

a formula corresponding to one that was given long ago by Clau- 
sius and Mossotti. Substituting in it Maxwell's value 

£ = ii\ (215) 

we find the relation 

7-fe^ = const. (216) 

In this way, however, the formula is only proved for the case of 
very slow vibrations, to which Maxwell's law (215) may be applied, 
whereas our former deduction shows that it holds for any value 
of n, i. e. for any particular kind of light we wish to consider. 

125. Let us now compare our formula with experimental results. 
Of course I can only mention a few of these. I shall first consider 
the changes in the refractivity of a gas produced by pressure, and 

1) L. Lorenz, Uber die Refraktionskonstante, Ann. Phys. Chem. 11 (1880), 
p. 70. 

Ijorentz, Theory of electrons. 10 



146 IV. PROPAGATION OF LIGHT IN A SYSTEM OP MOLECULES. 



in the second place the change in the refractivity that is brought 
about by the passage of a liquid to the gaseous state. In both 
cases I shall compare the results of our formula with those that can 
be deduced from the empirical formula (214). As to Laplace's law, 
we need no longer speak of it, because in aU cases it is much less 
satisfactory than either of the two other formulae. 

The refractive index of air up to high densities has recently 
been measured with considerable accuracy by Magri.^) Some of 
his results are contained in the following table, together with the 

values of ^ and -r^i~i;^ ■ 

3 Q (fi,' + 2)e 



Temperature 


Density 


Refractive index 


3 Q 


'''-' .10' 

(^^ + 2)9 


0" 


1 


1,0002929 


1953 


1953 


14,6 


14,84 


1,004338 


1919 


1947 


14,3 


42,13 


1,01241 


1964 


1959 


14,4 


69,24 


1,02044 


1968 


1961 


14,5 


96,16 


1,02842 


1970 


1961 


14,5 


123,04 


1,03633 


1969 


19.o6 


14,8 


149,53 


1,04421 


1971 


1956 


14,9 


176,27 


1,05213 


1972 


1953 



You see that with the formula (216) the agreement is somewhat 
better than with the empirical relation (214). 

The difference between the two comes out still more markedly, 
if we compare the refractive index of a vapour with the value we 
can deduce from that of the liquid by means of (214) or of (216). 
In the following small table, which relates to sodium light, the index 
of the liquid is given for 15", and that of the vapour for 0" and 
760 mm. This means that the observed values of fi have been reduced 
to the density which the vapour would have at 0" and under atmo- 
spheric pressure, if it followed the laws of Boyle and Gay-Lussac. 
The reduction can be made either by (214) or by (216), the two 
formulae being equally applicable to the small changes in question. 



Liquid 



Density 



Index of 
refraction 



Vapour 



Density 



Index of refraction 



Obs. 



Calculated 
by (214) 



Calculated 
by (216) 



Water 

Bisulphide of car- 
bon 

Ethyl ether . . . . 



0,9991 

1,2709 
0,7200 



1,3337 

1,6320 
1,3558 



0,000809 

0,00341 
0,00332 



1,000250 

1,00148 
1,00152 



1,000270 

1,00170 
1,00164 



1,000250 

1,00144 
1,00151 



1) L. Magri, Der Brechungsindex der Luft in seiner Beziehung zu ihrer 
Dichte, Phys. Zeitschr. 6 (1905), p. 629. 



REFRACTIVITY OF A MIXTURE. 147 

Other measurements which can be taken as a test for the two 
formulae are those of the indices of refraction of various bodies at 
different temperatures, or when submitted to different pressures. As 
a general rule, neither equation (214) nor (216) is found to repre- 
sent these measurements quite correctly, the disagreement between 
the observed values and the calculated ones being of the same order 
of magnitude in the two cases, and generally having opposite direc- 
tions. In most cases our formula leads to changes in the refractivity 
that are slightly greater than the observed ones; moreover, the devia- 
tions increase as one passes on to higher values of n. 

As to the cause of this disagreement, it must undoubtedly be 
looked for, partly in the fact that the term a in equation (201) is 
not exactly equal to -^, partly also in changes that take place in 
the interior of the particles when a body is heated or compressed. 
These changes can cause a variation in the value of the coefficients 
f and f. 

126. A problem closely connected with the preceding one is 
that of calculating the refractivity of a mixture from the refractivities 
of its constituents. Following the same line of thought that has led 
us to equation (212), but supposing the system to contain two or 
more sets of molecules mixed together, one finds the following for- 
mula'), in which r^, r^, ... are the values of 

^ (217) 



(^^ + 2)9 



for each of the mixed substances, taken separately, and m^, m^, . . . 
the masses of these substances contained in unit of mass of the mixture 

-^ff^r^ ^ '"''''' + '^2»-2 + • • • • (218) 

This equation is found to hold as a rough approximation for various 
liquid mixtures. The same may be said of a similar equation that 

is often used for calculating the value of 

127. It is very important that these formulae for mixtures can 
also serve in many cases for the purpose of calculating the refracti- 
vity of chemical compounds from that of the constituting elements. 
Let us consider a compound consisting of the elements s^, s^, ..., 
and let us denote hj Pi, P2, ■ ■ • their atomic weights, by g^, q^, ... 
the numbers of the different kinds of atoms in a molecule, and by 

P = ilPl + Q2P2-\ 

1) Note 58. 

10* 



148 IV. PROPAGATION OP LIGHT IN A SYSTEM OP MOLECULES. 

the molecular weight of the compound. Then, the amounts of s^, 
s^, ... in unit of mass will be 



and (218) becomes 

^WT2Ji = QiPi^'i + ^iP2^2 + ■■■■ (219) 

Hence, if for each element we call the product of the constant (217) 
by its atomic weight p the refraction equivalent, and if we under- 
stand by the refraction equivalent of the compound the product of 
the value of (217) relating to it by the molecular weight P, we are 
led to the simple rule that, in order to find the refraction equivalent 
of the compound, we have only to multiply the refraction equivalent 
of each element by the number of its atoms in the molecule, and to 
add the results. A large number of physicists and chemists who 
have determined the refractivities of many compounds, especially of 
organic ones, have found the rule to be approximately correct. 

128. The general meaning of this result will be obvious. When 
we find that some quantity which determines the refractivity of a 
compoimd is made up of a number of parts, each of which belongs 
to one of the elements, we may conclude that, in the propagation of 
light, each element exerts an influence of its own, which is not disturb- 
ed by the influence of the other elements. In the terms of our theory, 
this amounts to saying that the electric vibrations going on in a beam 
of light, in so far as they take place in the ponderable matter, have 
their seat in the separate atoms, the motions in one atom being more 
or less independent of those in the other atoms of the same molecule. 

We may suppose, for example, that each atom contains one 
movable electron, which, after a displacement from its position of 
equilibrium, is puUed back towards it by an elastic force having its 
origin in the atom itself, and determined therefore by the properties 
of the atom. If we take this view, it is easy so to change the 
equations for the propagation of light that they can be applied to 
a system of polyatomic molecules. 

129. Let us distinguish the quantities relating to the separate 
atoms of a molecule by the indices 1, 2, . . ., Jc. Let e^, gg, . . . 
be the charges of the movable electrons contained in the first, the 
second atom etc., m-,, m.^, ... their masses, g^, i]^, g,, I2, 1;., g^; . . 
the components of their displacements from their positions of equili- 
brium, /i, f.^, . . the coefficients determining the intensities of the 
elastic forces. Then, if the resistances are left out of account, and 



liEFRACTrVITY OF A CHEMICAL COMPOUND. 



149 



if there is no external magnetic field, we shall have for each molecule, 
not one set of equations of motion of the form 



dt" 



etc.. 



(which is got from (198) if we put ^ = 0, § = 0), but h sets of 
this form: 



m 



m, 



1 dt''^ = ^1 <^^^ + «PJ - /i ^1 , etc. 
8 dt^ 



(220) 



e2(E^ + aPJ-/'2l2, etc. 

etc. 

The total electric polarization P of the body wiU now be the sum 
of the electric moments due to the separate atoms; its components are 

P,j = N(e,7], + e,^,+ -..), (221) 

For a determinate value of the frequency n we can deduce 
from (220), (221) and (192) the relation between E and D. Combi- 
ning it with the equations (193) and (194), one finds the following 
formula, corresponding to (212), but more general than it, for the 
index of refraction f/.^). 



Ne, 



+ 



Ne, 



+ 



(222) 



It is thus seen that, according to our new assumptions, the value of 

2,„ remains proportional to N, and therefore to the density of the 

body. Moreover, if we denote by ftj, /Xg, ... the refractive indices 
for the cases that unit volume of our system contains only N atoms 
of the group 1, or N atoms of the group 2, etc., we have 



Consequently, (222) may be written 



Sif.-m.n")' 



^^-1 



ft' 






which is but another form of the relation (219). 

130. I need hardly observe that the assumptions we have made 
are at best rough approximations to the true state of things. We 



1) Note 69. 



150 IV. PROPAGATION OP LIGHT IN A SYSTEM OP MOLECULES. 

haye supposed, tlie elastic force by wliicli the movable electron of an 
atom is puUed towards its position of equilibi-ium, to arise from 
actions wliicli are confined to the atom itself. Now^ there is at all 
events an interaction of an electric nature between neighboring 
atoms, precisely on account of the displacements of their electrons; 
there may also be other interactions about whose nature we are as 
yet entirely in the dark. On these grounds we must expect greater 
or smaller deviations from the law of the refraction equivalent, 
deviations from which one may one day be able to draw some con- 
clusions concerning the structure of a molecule. 

One important result in this direction was already obtained by 
Briihl.^) He found that a double chemical binding between two atoms 
has a striking influence on the refractivity, which can be taken into 
account, if, in the formula (218), we add a term of proper magnitude 
for each double bond. 

Like many other facts, this shows that our theory of the 
propagation of light in ponderable bodies is to be considered as 
rather tentative. I must repeat however that, undoubtedly, the actions 
going on in the separate atoms must be, to a large extent, mutually 
independent. If they were not, and if, on the contrary, the elastic 
force acting on an electron ought to be attributed, not to the atom 
to which it belongs, but to the molecule as a whole, the refractive 
index of a compound body would be principally determined by the 
connexions between the atoms, and not, at it is, by their individual 
properties. 

131. At the point which we have now reached, it is interesting, 
once more to return to the theory of the dispersion of light, and to 
ask what the general formula (222) can teach us about it. To begin 

with, it may be observed that, if s is the value of H-i — , we shall 

have 

„2_ 1 + 2*' 



) 



from which it is readily seen that, when s continually changes from — ^ 
to 1, [i^ increases from to oo. If s remains confined to this inter- 
val, as I shaU for the moment suppose it to do, /i changes in the 
same direction as s, having the value 1 for s = 0. 

The latter case occurs for iV=0, i. e. when there are no ponderable 
particles at aU, so that the propagation takes place in the ether alone. 
This state of things is altered by the presence of the electrons, to 

3) See, for instance, J. W. Briihl, The development of spectro-chemistry, 
Pioc. Royal Institution, 18, 1 (1906), p. 122. 



DISPERSION OF LIGHT. 



151 



which the different terms on the right-hand side of (2-22) relate. 
Now, each of these electrons has a definite period of its own, in 
which it can perform its free vibrations. If the frequencies of these 
are n^, n^, etc., we shall have 



and 






n.' = 



,(i^ + 2 3j»i(w,' 



+ 



Ne, 



SjKj {n^' — n') 



^^ + 



(223) 




The influence of an atom is thus seen to depend on whether the fre- 
quency of the rays for which we wish to determine [i, lies below or 
above the frequency of the free vibrations. Each group of electrons 

tends to raise the value of ^ and consequently that of ft, for aU 

frequencies below its own, and to lower the refractive index for all 
higher frequencies. 

As a function of n, each term of (223) can be graphically 
represented by a curve of the form shown in Fig. 5, in which OP 
corresponds to n^, Wj, . . ., as the case may 
be, and we shall obtain the curve for 

' ; I ^ by taking the algebraic sum of the 

ordinates in the individual curves L^ , L.^ . etc. 

The form of the resultant line wiU 
be determined by the values of n^, n^, etc. 
or, as one may say, by the position in — 
the spectrum of the lines that would be ^ 
produced by the free vibrations of the 
electrons, and which we may provisionally 
call the spectral lines of the body. If, as 
we go from left to right, we pass one of 
these lines, the ordinate of the correspon- Fig. 5. 

ding curve suddenly jumps from + oo 

to — oo. Of course aU these discontinuities are repeated in the re- 
sultant dispersion curve, and near each of the values n^, n^, . . . of the 
frequency there wiU be a portion of the curve, in which s first 
changes from -f 1 to -}- oo, and then from — oo to — ^. It may 
be assumed that these portions, which I shall call the discontinuous 
parts of the curve, have a breadth that is very smaR in comparison 
with the remaining parts, of which I shall speak as the continuous ones. 

Since all the curves L^, L^, etc. rise from left to right, it is 
clear that each continuous portion of the resultant curve must pre- 
sent the same feature. This agrees with the dispersion as it exists 
in all transparent substances. 




152 IV- PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES. 

The question as to whether, for a definite value of the frequency, 
the index of refraction is greater or less than 1, depends on circum- 
stances. If all the spectral lines of the body lie in the ultra-violet, 
the refractive index will be larger than 1 throughout the infra-red 
and the visible spectrum. It may remain so in the visible part, even 
if there are one or more infra-red lines, provided only there be also 
lines in the ultra-violet, whose influence in raising the refractive index 
predominates over the opposite influence of the lines in the infra-red. 
At aU. e rents, the dispersion of light observed in all transparent 
bodies requires for its explanation the existence of one or more lines 
in the ultra-violet. 

132. We could now enter upon a comparison of our dispersion 
formula with the measurements of the indices of refraction, but I 
shall omit this, because we must not attach too much importance to 
the particular form which we have found for the equation. By 
slightly altering the assumptions on which it is based, it would be 
possible to find an equation of a somewhat different form, though 
agreeing with (223) in its main features. There is, however, one 
consequence resulting from the preceding theory, to which I should like 
to draw your attention. If the frequency n is made to increase in- 
definitely, an the terms on the right-hand side of (223) approach 
the limit 0; hence, for very high frequencies, we shall ultimately 
have s = 0, and n = 1, the reason being simply, that the electrons 
cannot follow electric forces alternating with a frequency far above 
that of their free vibrations. The remark is important, because it ex- 
plains the fact that the Rontgen rays do not suffer any refraction 
when they enter a ponderable body. These rays, though not con- 
stituted by regular vibrations, are in all probability produced by a 
very rapid succession of electromagnetic disturbances of extremely 
short duration. 

133. Thus far we have only spoken of the continuous parts of 
the dispersion curve. In each of its discontinuous portions, as we 
have defined them, the right-hand side of (223) has values ranging 
from -t- 1 to + <x>, and from — oo to — |. These values lead to 
negative values of [i^, and to imaginary values of fi itself, indicating 
thereby that waves of the corresponding frequencies cannot be pro- 
pagated by the body in the same way as those whose wave-length 
corresponds to a point in one of the continuous parts of the curve. 

We need not however any further discuss the meaning of our 
formulae in this case, because, for frequencies very near n^, n^, . . . 
the resistance to the vibrations, and the absorption due to it may no 
longer be neglected. We must therefore now take up the subject 



ABSORPTION OF LIGHT. 153 

of the absorption of light. Not to complicate matters too much, I 
shaU do so on the assumption, which we originally started from, that 
each particle contains a single movable electron. 

If, in the equations (204), the resistance coefficient /3 has a 
certain value, and if there is no external magnetic field, we may write 

E = (« + i/3)P. 
This gives 

■J-^ + P^l^ + c^.l^- 

On the other hand, the equations deduced in § 121 from (193) and 
(194) remain unchanged, so that we find, instead of (210), 

'^V=l + ^- (224) 

The constant q in the expression (209) now becomes a complex 
quantity. It is convenient to put it in the form 

« = i-4' (225) 

V and h being real. Then^ (209) becomes 

— kz-\-in\t 1 

E \ '/, 

and if, in order to find the expressions for the vibrations, we take 
the real parts of the complex quantities by which the dependent 
variables E^ etc. have first been represented, we are led to expres- 
sions of the form 

E-'" cos n{t — -^+p^, (226) 

where p is a, constant. The meaning of the first factor is, that the 
amplitude of the vibrations is continually decreasing as we proceed 
in the direction of propagation. The light is absorbed to a degree 
depending on the coefficient fc, which I shaU call the index of ab- 
sorption. On the other hand, the second factor in (226) shows that 

V is the velocity with which the phase of the vibrations is propagated ; 

the ratio — , for which I shall write ft, is therefore properly caUed 

the refractive index. 

Substituting (225) in (224), and separating the real and the 
imaginary parts, one finds the following equations for the deter- 
mination of V (or ,u) and h^') 



^ + '7^ + 1;^+^' (227) 



-§^ ' oj^-f (3 



2f.^ = ]/l 
2^'=l/l^lSf-,^-l- (228) 



1) Note 60. 



154 IV. PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES. 

134. The discussion of these formulae, which in general would 
be rather complicated, can be considerably simplified by the assump- 
tion that /3 is much larger than 1. 

This is true in the majority of cases, because in nearly aU bodies 
the absorption in a layer whose thickness is equal to one wave-length 

in air, i. e. to , is very feeble, even for those frequencies for 

which the absorption is strongest. According to (226) the amplitude 
diminishes in the ratio of 1 to 

£ n , 

while the beam travels over a distance . Therefore, for the 

bodies in question, 

clc 



must be a very small number. Now, if we consider the particular 
frequency for which a = 0, (228) becomes 



2?' = l/i + i-i- 



If this is to be very smaU, /J must be a large number. 

Availing ourselves of this circumstance, we find the following 
approximate equations ^) 

For a given value of /3 the fraction 



has its greatest value -g- for a = 0. For « = + 13 it has sunk to half 

this maximum value, and for « = + vj3 to , „ , — r-:^. If we under- 

stand by v a moderate number (say 3 or 6) the absorption can be 
said to be very feeble, in comparison with its maximum intensity, 
for values of a beyond the interval extending from — v/J to -(- vj3. 

135. These different cases succeed each other as we pass 
through the spectrum, and even, notwithstanding the high value we 
have ascribed to /3, the transition from ~ v^ to + v/S can take 
place in a very narrow part of it. If we suppose this to be the 



1) Note 61. 



BEEADTH AND INTENSITY OF AN ABSORPTION BAND. 155 

case, the factor g- in (230), and the factor n in (202) may be con- 
sidered as constants. Moreover, in virtue of (201), if w^' is the 
frequency for which « = 0, we may write for any other value of a 
in the interval in question 

a = _ 2 m'<(w - V)- (231) 

I have written n^' for the frequency corresponding to a = 0, because 
its value 

differs from the frequency 

of the spectral line of a detached molecule of which we have formerly 
spoken. It is only when we may neglect the coefficient a, that the 
two may be considered as identical. 

The phenomena which the system of molecules produces in the 
spectrum of a beam of white light which is sent across it, are as 
follows. There is an absorption band in which the place of greatest 
darkness corresponds to 

n = Wq'. 

The distribution of light is symmetrical on both sides of this point. 
As the band has no sharp borders, we cannot ascribe to it a definite 
breadth; we can, however, say that it is seen between the places 
where « = — v^ and « = + v/3, v being a number of moderate 
magnitude. Measured by a difference of frequencies, half the width 
can therefore be represented by 

as is seen from (231). 

We may add an interesting remark about the intensity of the 
absorption. The maximum value of the index of absorption is found 

to be 

n 

and the formulae (202j, (199) and (207) show therefore that the maxi- 
mum is the larger, the smaller the resistance, or the longer the time r 
durinc which the vibrations of the electrons remain undisturbed. This 
result, strange at first sight, can be understood, if we take into con- 
sideration that the vibrations which are set up in 'a particle by optical 
resonance, so to say, with the incident Hght, wiU be sooner or later 
converted into an irregular heat motion. It may very well be, that 



156 IV- PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES. 

the total quantity of heat developed per unit of time is larger when 
vibratory energy is stored up during a long time, and then suddenly 
converted, than in a case in which the disturbances take place at 
shorter intervals. 

In another sense, however, the absorption may be said to be 
intensified by an increase of the resistance g, or by a shortening of 
the time x. Not only wiU a change of this kind enlarge the breadth 
of the absorption line; it will also heighten the total absorption, 
i. e. the amount of energy, all wave-lengths taken together, that is 
taken up from an incident beam.-') 

As a general rule, observation really shows that narrow absorp- 
tion bands are more intense in the middle than broad ones. 

136. In Fig. 6 the curve FGH represents the index of ab- 
sorption as a function of the frequency. The other curve ABODE 
relates to the index of refraction; it corresponds to the formula (229). 

The index ^, which is 1 at large distances 
on either side of the point P, rises to a 
maximum QB, and then sinks to a mini- 
mum i?jD. The place of the maximum 
is determined by a = j3, or 




n = ' 



2 m n^ ' 
that of the minimum by « = 

w = Wo + s— ' — ' , 



fi, or 



the corresponding values of (i being 



1 + ^ 



and 1 — 



4^ 



The maximum and the minimum are found at points of the spectrum 
where the index of absorption has half its maximum value. 

In the line ABODE one wiU have recognized already the well 
known curve for the so-caUed anomalous dispersion. I must add that, 
if we had supposed, as we did in § 128, the particles of the system 
to be composed of a certain number of atoms, each containing a 
movable electron, and if we had assumed a resistance for every 
electron, we should finally have found a dispersion curve in which 
a part of the form ABODE repeats itself in the neighborhood 
of each free vibration. These parts would take the place of the 
discontinuous portions that would exist in the curve for the func- 
tion (223). 



1) Note 62. 



PfiOPAGATION ALONG THE LINES OF FORCE. ]57 

137. The effect of an external magnetic field on the propagation 
of light in the direction of the lines of force can be examined by- 
calculations much resembling the preceding ones. We have again to 
use the equations (192), (193) and (194), but we must now combine 
them with the formulae (204). Since, in these latter, the force § 
has been supposed to have the direction of OZ, a beam of light 
traveRing along the lines of force can be represented by expressions 
containing the factor (209). We are again led to the equation 

to which we must now add the corresponding formula 

which there was no occasion to consider in the preceding case. Using 
(192), we find 

P^=(cV-l)E,, P^=(cV-l)E„ 

and the first and the second of the equations (204) become 

(232) 



1 



c'2 



■3^-(« + i^)|P,= + tyP,, 



showing that 

P,= + iP,. (233) 

Thus, there are two solutions, corresponding to the double sign. In 
order to find out what they mean, we must remember that, if two 
variable quantities are given by the real parts of 

a£«(«'+p) and a^£»(««+p+2'^»)^ (234) 

i. e. if they are represented by 

a Q.o^{nt -{■ p) and ar cos(w< -f-p + 23ts), 

the number r determines the ratio between the maximum values or 
amplitudes, whereas s is the phase-difference expressed in periods. 
Since rs^"*', the ratio between the expressions (234), becomes + i, 

when we take 

r=\, s = + i-, 

equation (233) shows that P^, and P have equal amplitudes and that, 
between their variations, there is a phase-difference of a quarter period. 
The same may be said of the displacements | and 7] of one of the 
movable electrons, these quantities being proportional to P^ and P^^. 
We can conclude from this that each electron moves with constant 
velocity in a circle, whose plane is perpendicular to OZ, the motion 
having one direction in the solution corresponding to the upper sign, 
and the opposite direction in the other solution. 



158 IV. PROPAGATION OF LIGHT IN A SfSTEM OP MOLECULES. 

Similarly, the vector P has a uniform rotation in a plane at 
right angles to OZ, and the same is true of the vectors E and D. 
Each of our solutions therefore represents a beam of circularly pola- 
rized light, and it is easily seen that, when the real part of ^ has the 
positive sign (so that the propagation takes place in the direction of 
the positive s) the upper signs in our formulae relate to light whose 
circular polarization is right-handed, and the under signs to a left- 
handed polarization. 

Now, if we substitute the value (233) in either of the formulae 
(232), we obtain the following condition for the coefficient q: 

«V = 1+^^, (235) 

from which, if we introduce the value (225), the index of absorption 
and the velocity v, or the index of refraction fx, can be calculated. 

138. It is not necessary to write down the expressions for these 
quantities. Comparing (235) with our former equation (224), we 
immediately see that the only difference between the two is, that a 
has been replaced by a + j'. Now, in a narrow part of the spectrum, 
y may be considered as a constant. Therefore, if we use right-handed 
circularly polarized light, the values of li and ft which correspond to a 
definite value of a must be the same as those which we had for the 
value tt -\- y in the absence of a magnetic field. On account of the 
relation (231), we can express the same thing by saying that, 
in the neighborhood of the frequency n^, the values of ju, and Ic 
for a frequency n are, under the influence of the magnetic force §, 
what they would be without this influence for the frequency 



im'nJ ' 



The absorption curve for a right-hand ray is therefore obtained by 
shifting the curve FGH of Fig. 6 over a distance 



— , (236) 



2 m w^ 

the displacement being in the direction of the increasing frequencies, 
when this expression is positive, and in the opposite direction, when 
it is negative. For the left-hand ray we find an equal displacement 
in the opposite direction. 

It is clear that the inverse Zeeman effect is hereby explained. 
If a beam of unpolarized light, which we can decompose into a right 
and a left-handed beam, is sent through the flame, we shaR get in its 
spectrum both the absorptions of which we have spoken. If the 
distance (236) is large enough in comparison with the breadth of the 
region of absorption, we shall see a division of the dark band into 



PROPAGATION AT RIGHT ANGLES TO THE LINES OP FORCE. 159 

two components. It is especially interesting that the disphicement 
(236), for which by (203) and (199) we may write 

cNe ' Ne^ '^ 2toc ' 

exactly agrees with the value we found in the elementary theory of 
the direct Zeeman effect. Our result also accords with our former 
one as to the direction of the effect. When we examined the emission 
in the direction of the magnetic lines of force, we found that the 
light of the component of highest frequency is left-handed, if e is 
negative. Our present investigation shows that for light of this kind, 
if e is again supposed to be negative, the absorption band is shifted 
towards the side of the greater frequencies. 

139. The propagation of light in a direction perpendicular to 
the lines of force can be treated in a similar way. If the axis of x 
is laid in the direction of propagation, the axis of z being, as before, 
in the direction of the field, and if we assume that the expressions 
for the components of E, D, P and H contain the factor 






x + in tt 1 



k win again be the index of absorption, and v the velocity of pro- 
pagation. 

Now, it is immediately seen that these quantities are not in the 
least affected by the magnetic field, if the electric vibrations of the 
beam are parallel to the lines of force, for in this case we have only 
to combine the last of the equations (204) with the relations 



D, = E, + P 



z> 



dx ~ e dt ' dx e dt ' 

which are included in (192) — (194). Since none of these formulae 
contains the external force §, we may at once conclude that the 
magnetic field has no influence on the electric vibrations along the 
lines of force. 

As to vibrations at right angles to these lines, I must point out 
a curious circumstance. The variable vectors being periodic functions 
of the time, and depending only on the one coordinate x, the con- 
dition 

div b = 0, 

which foUows from (193), requires that 

D, = 0. (237) 



160 IV. PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES. 

We can express this by saying that the electric vibrations have 
no longitudinal component, meaning by „electric vibrations" the 
periodic changes of the vector D. But our result by no means ex- 
cludes values of E^ and P^ different from 0, so that, if the denomi- 
nation of electric vibrations is applied to the fluctuations of the 
electric force E, or of the polarization P, the vibrations cannot be 
said to be transversal. 

The formula (237) is very important for the solution of our 
problem. Writing it in the form 

E. + P. = 0, 

we can deduce from (204), combined with (192), 



p = ^y ., P , (238) 

{l + a + ipf-vl 



V 



The condition 

J_ 3'Dy ^ g^Ey 

which follows from (193) and (194), will therefore be fulfilled if 

\v n) c^ ■ (l+a + i^)(a + «(3) — y'' ^ ^ 

This is the equation by which the velocity of propagation and the 
index of absorption can be calculated. At the same time, the ratio 
between P^ and P may be taken from (238). If. for this ratio we 
find the complex value rs^"'' (§ 137), so that 

P = rF^"'" P 

the amplitudes of P^ and P are as r to 1, and there is a phase- 
difference measured by s between the periodic changes of the two 
components. The quantities r and s also determine the ratio of the 
amplitudes and the difference of phase for the vibrations along OX. 
and Y into which the motion of one of the movable electrons can 
be decomposed. Generally speaking, in the case now under considera- 
tion, the path described by each electron is an ellipse in a plane 
perpendicular to the lines of force. Since r and s vary with the 
frequency, the form and orientation of the ellipse will depend on the 
kind of light by which the flame is traversed. 

140. In order to find the position of the absorption lines in 
the spectrum, we should have to determine k by means of the equa- 
tion (239), and to seek the values of the frequency which make h 
a maximum. If the denominator of the last fraction in (239) is 
divested of imaginaries, the equation takes the form 



TRANSVERSE ZEEMAN-EPFECT. igl 



and we get 



\v n) c^ 



Q 



^^'=27^--^=^-- (240) 

The general discussion of this result leads to formulae of such com- 
plexity that they can hardly be handled. Fortunately, in the cases 
we shall have to consider, the frequencies for which Ic is a maximum, 
may he found with sufficient accuracy by making the denominator Q 
a minimum. Moreover, in doing so, we may again consider a as the 
only variable quantity, the quantities /3 and y not varying perceptibly 
in the narrow part of the spectrum with which we are concerned. 
Now, the denominator may be written in the form') 

from which it immediately appears that the values in question are 
given by 

I shall suppose that 

so that the equation has two real roots 

« = - i + Vf^fi'^+~i- (241) 

Corresponding to these, one finds 

^ = 4/3V', ^ = ^/3(l + 4/ + 1/47^^4^+1), (;) = ^2(i + 4y^), 

(242) 



»y 



i±V7'-fi' + i + i§ 



141. These results take a very simple form, when, as is generally 
the case (§ 134), j3 is great in comparison with 1, and the mag- 
netic field is so strong that, for light travelling along the lines of 
force, the components of the original absorption line are separated 
to a distance greatly surpassing their breadth. This requires (§ 138) 
that y be stiU many times greater than /3. Instead of (241) we may 
therefore write approximately 

a = ±y, (243) 

which shows that there are two absorption lines, exactly at the points 
of the spectrum where we had lines when the light had the direction 
of the lines of force, i. e. in the positions which the elementary 
theory of the direct Ze em an -effect might lead us to expect. 



1) Note 63. 

Lorentz, Theory of electrons. 11 



162 IV. PROPAGATION OP LIGHT IN A SYSTEM OP MOLBCCJLES. 

In calculating the index of absorption we may now replace B 
by 2/3y2 and Q by 4/3^'- Since 

VA^ + £2 - A = --=^L , 

^ Ya' + b' + a' 

we find for both lines 

Ic = -- (244) 

exactly one half of the index of absorption corresponding to a = 
in the absence of a magnetic field. 

Finally, the expression (242) has the value + i, so that we may 
conclude as follows. 

If the absorbing body is traversed, in the direction of OX, by 
a beam of light whose electric vibrations were originally parallel 
to Y, the vibrations are absorbed to an amount determined by (244), 
when the frequency has either of the values given by (243). The elec- 
trons in the molecules will describe circles in planes parallel to OX 
and Y, the direction of their motion corresponding to that of the 
magnetic force when a = + y, and not corresponding to it when 
« = — 2'. 

It should be noticed that, in the case treated in § 138, in 
accordance with our present result, we found the maximal absorption 
at the point a = -\- y, if the circular motion had the former, and 
at the place « = — y, if it had the latter of the directions just named. 

142. Voigt has drawn from his equations another very re- 
markable conclusion. In general, for a beam of light travelling at 
right angles to the lines of force, and consisting of electric vibrations 
perpendicular to these lines, the two absorption bands which we get 
instead of the single original one, are neither equally distant from 
the position of the latter, nor equally strong, as the components of 
the doublet observed in the direction of the field invariably are. 
This follows immediately from the circumstance that the functions 
A, B and Q contain not only even, but also odd powers of a, so 
that the phenomena are not symmetrical on both sides of the point 
in the spectrum where a == 0. 

In some experiments undertaken by Zee man for the purpose 
of testing these predictions, a very slight want of symmetry was in- 
deed detected. If this is really the dissymmetry to which Voio-t 
was led by his calculations, the phenomenon is highly interesting, as 
we can deduce from it that the gaSeous body in which it occurs 
exerts what we may call a metallic absorption in the middle of the 
band. Indeed, the peculiarity to which Voigt called attention, can 
make itself felt only in case the coefficient /3 is not much oreater than 



ROTATION OF THE PLANE OF POLARIZATION. 163 

unity, and this leads to an absorption -which is very sensible even for 
a thickness equal to a wave-length (§ 134).^) 

143. I must now call your attention to the intimate connexion 
between the Zeeman-effect and the rotation of the plane of polari- 
zation that was discovered by Faraday. Reverting to the case of 
a propagation along the lines of force, we can start from our former 
result (§§ 137, 138), that the simplest solutions of our system of 
equations are those which represent either a right-handed or a left- 
handed circularly polarized beam, and that the formulae for these 
two cases are obtained, if, in the equations holding in the absence 
of a magnetic field, we replace a either hj a -\- y or hj a — y. This 
is true, not only for the formulae giving the coefficient of absorption, 
but also for those which determine the velocity of propagation. 
Hence, if this velocity is denoted by tij for a left-handed, and by 
v^ for a right-handed ray, we shall have (cf. equation (229)), 
1 ^ 1 _l <^ — Y 



», c ' 2c[(a — y)=i + p*]' 
1 _£_^ « + )- 



v,_ c T2c[(a + y)^ + /3»] 
For a definite frequency n, these values are unequal. So are also 
the corresponding values of the coefficient of absorption, so that, 
under the influence of a magnetic field, the system has a different 
degree of transparency for the two kinds of circularly polarized light. 
For the sake of simplicity, however, we shall now leave out of con- 
sideration this latter difference, and speak only of the phenomena 
that are caused by the difference in the velocities of propagation. 
You know that in every case in which these are unequal for the two 
kinds of circularly polarized light, a beam with a rectilinear polari- 
zation will have its plane of polarization turned as it travels onward. 
The angle of rotation per unit of length is given by 

- = Y-(i-^)' (245) 

or in our case by 

n r a-\ -y a — y "] /ni^^s, 

<" = I^ LTcT-T -yf+J^ - («-r)^ + H ' ^ ' 

The sense of the rotation depends on the algebraical sign of 

this expression. When § is positive, i. e. when the magnetic force 

has the direction of the beam of light, a positive value of m means a 

rotation whose direction corresponds to that of the magnetic force. 

The general features of the phenomenon, as it depends on the 

frequency, come out most clearly if we avail ourselves of a graphical 

representation. In Fig. 6 we drew a curve giving the index of re- 

1) See Note 64. 

11* 



164 IV. PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES. 



fraction as a function of the frequency, and showing how it changes 
as we go through the spectrum from left to right. This curve, which 
relates to the body not subjected to a magnetic force, may also be 

taken to represent the values of — ■ Now, if there is a magnetic 

field, the curves for — and — are obtained by simply shifting that 
of Fig. 6 towards the left or towards the right over an interval 



equal to 



2mc 



(cf. § 138). In this way we get the two curves 




A-^B^D^E^ and A^B^D^E^ of Fig. 7, and these immediately give us 

an idea of the angle of rotation o, 
because, as (245) teUs us, this angle 
is proportional to the algebraic diffe- 
rence between corresponding ordinates. 
It can therefore be represented by 
the line BR. 

Two interesting results become 
apparent by this construction. The 
first is, that in the narrow part of 
the spectrum close to the original 
absorption line, the rotation of the 
plane of polarization twice changes 
its sign; the second, that, on account 

of the high values of ^ or — which 

are found at some places, the angle 
of rotation can also attain a rather 
great value. 
Macaluso and Corbino-"^), who were the first to examine this 
phenomenon in the case of a sodium flame, observed rotations as 
great as 270". The results of their experiments could immediately 
be explained by the theory which Voigt had already developed. 
Some years later, Zeeman^) and Hallo^) made a very careful ex- 
perimental study of the phenomenon, and again found a satisfactory 
agreement with Voigt's theory. 

144. The Faraday -effect had been known for a long time, and 
the only thing in the above results apt to cause astonishment, was, 
that a rotation much greater than had ever been observed in trans- 
parent bodies, should be produced in a sodium flame. An other 
magneto-optic effect that was predicted by Voigt, is an entirely new 

1) D. Macaluso and 0. M. Corbino, Comptes rendus 127 (1898), p. 548. 

2) P. Zeeman, AmsterdamProc. 6(1902),p. 41; Arch. n(5erL (2)7 (1902), p. 4G5. 

3) J. J. Hallo, Arch. n(3erl. (2) 10 (1905), p. 148. 



MAGNETIC DOUBLE REFRACTION. 165 

one. It consists in a double refraction that is observed when a body such 
as we have considered in this chapter, is traversed by a beam of 
light at right angles to the lines of force. For such a beam we 
have to distinguish between the electric vibrations perpendicular and 
parallel to the lines of force. For the former, the velocity of propa- 
gation is given by the equation (239), for the latter by (224) and 
(225), or as we may also say, by (239), if in this latter formula we 
put y = 0. The difference between the two values is what was 
meant when I spoke just now of a double refraction. It can be 
calculated by our formulae as soon as a, /3, y are known, but I shall 
not lose time in these calculations. I shall only observe that the 
effect remains the same when the field is reversed; this follows at 
once from (230), because this equation contains only the square of y, 
and therefore the square of !Q. 

Voigt and Wiechert have experimentally verified these predic- 
tions, and Geest^) has carefully measured the magnetic double refrac- 
tion in a sodium flame. 

145. Availing ourselves of the theory that has been set forth 
in this chapter, we can draw from experimental data certain interesting 
conclusions concerning the absorbing (or radiating) particles. Some 
measurements enable us to calculate the relative values of the three 
quantities a, j3, y, whereas others can serve for the determination of 
their absolute values. 

Thus, if we have measured the distance between the middle com- 
ponent of Zeeman's triplet and the outer ones, we know that for 
the frequency n belonging to one of these latter, a and y have equal 
values. Replacing y by its value (203), and a by (231), in which 
we shall now neglect the difference between n^' and Wq (§ 135), so that 

a = 2 m «o (wo - w) = ^-^ -' , (247) 

the equality leads us back to our old equation 

" 2mc ' 

bv means of which we can determine the ratio — 

TO 

The ratio between a and /3 could be found if quantitative deter- 
minations of the absorption, in the ordinary case in which there is 
no magnetic field, were at our disposal. If, for example, we knew 
that at a certain point in an absorption band the index of absorption 
]c is X times smaller than at the middle of the band, the ratio v 
between a and /3 could be found (§ 134) by means of the formula 

x^l+v'. (248) 

1) J. Geest, Arch, n^erl. (2) 10 (1905), p. 291. 



166 IV. PROPAGATION OF LIGHT IN A SYSTEM OF MOLECULES. 

The distribution of intensity has been determined by bolometric or 
similar measurements for the broad bands that are produced by such 
gases as carbonic acid, but we cannot tell what it is in the ■ narrow 
bands observed in the case of a sodium flame for instance. All we 
can then do, is to form an estimate of the ratio v between a and /3 
for the border of the band. If we assume, for example, that here n 
is equal to 10 or 20, we can calculate v from the relation (248), 
and, substituting this value in the equation 

a = v^, 

for which, on account of (247), (202) and (199), we may write 

2ni(ng — n) = vff, 
we find 

2 m (w„ — n) 



This formula takes an interesting form if we use the relation (207). 
It then becomes 



showing that the time during which the vibrations in a particle go 
on uadisturbed may be deduced from the breadth of the band. 

In Hallo's experiments the breadth of the i)-lines was about 
one Angstrom unit, from which I infer that the value of r lies 
between 12-10-^^ and 24- 10^^^ sec. The first number is got by 
putting V = 3 (x = 10), the second by taking v = 6 (x = 37). As 
the interval between two successive encounters of a molecule is pro- 
bably of the order 10""* sec, we see that t comes out somewhat 
smaller than this interval, as was already mentioned in § 120. 

After having found the ratios between k, ^, y, we can try to 
evaluate the absolute values of these coefficients. For this purpose, 
we could use the absolute value of the coefficient of absorption, if 
it were but known. We can also avail ourselves, as Hallo and 
Geest have pointed out, of the rotation of the plane of polarization, 
or of the magnetic double refraction. If the ratios between «, /3, y 
are given, the three quantities may be deduced from the formula (246), 
or from the difl^erence between the value of v given by (239), and 
the corresponding value for ^ = 0. 

Now, when a. is known for a certain point in the spectrum in the 
neighbourhood of the point w„, i. e. when we know the value of (247), 

and if further we introduce the values of - and e, we can draw a 

conclusion as to the number of absorbing (or radiating) particles per 



NUMBER OF ABSORBING PARTICLES. 167 

unit of volume. In this way one iinds for the sodium flame used 
by Hallo 

N=4:- IQi^ 

corresponding to a density of the sodium vapour of about 10~^- In 
all probability this value is very much smaller than the density of 
the vapour actually present in the flame, a difference that must per- 
haps be explained by supposing that only those particles which are 
in a peculiar state, a small portion of the whole number, play a 
part in the phenomenon of absorption. 

I need scarcely add that all these conclusions must be regarded 
with some diffidence. To say the truth, the theory of the absorption 
and emission of light by ponderable bodies is yet in its infancy. If 
we should feel inclined to think better of it, and to be satisfied with 
the results already obtained, our illusion wiU soon be dispelled, when we 
think a moment of Wood's investigations about the optical properties 
of sodium vapour, which show that a molecule of this substance must 
have a wonderful complexity, or of the shifting of the spectral 
lines by pressure that was discovered by Humphreys and Mohler, 
and which the theory in its present state is hardly able to account for.-^) 



1) Note 64. 



CHAPTER V. 

OPTICAL PHENOMENA IN MOVING BODIES. 

146. The electromagnetic and optical phenomena in systems 
having a motion of translation^ as all terrestrial bodies have by 
the annual motion of the earth, are of much interest, not only in 
themselves, but also because they furnish us with means of testing 
the different theories of electricity that have been proposed. The 
theory of electrons has even been developed partly vrith a view to 
these phenomena. For these reasons I shall devote the last part of 
my lectures to some questions relating to the propagation of light 
in moving bodies and, in the first place, to the astronomical aberra- 
tion of light. 

Before I go into some details concerning the attempts that have 
been made to explain this influence of the earth's motion on the 
apparent position of the stars, it will be well to set forth a general 
mode of reasoning that can be used in problems concerning the 
propagation of waves and rays of light. It consists in the application 
of Huygens's well known principle. 

We shall consider a transparent medium of any kind we like, 
moving in one way or another, and we shall refer this motion and 
the propagation of light in the medium to three rectangular axes of 
coordinates, which we may conceive as likewise moving. We shall 

suppose our diagrams, which are to re- 
present the successive positions of waves 
of light, to be rigidly fixed to the axes, 
so that these have an invariable position 
in the diagrams. 

Let 6 (Pig. 8) be a wave -front in 
the position it occupies at the time t, 
and let us seek to determine the po- 
sition (j' which it win have reached 
after an infinitely short time dt. For 
this purpose we must regard each point P of (J as a centre of 
vibration, and construct around it the elementary wave that is formed 




Fig. 8. 



APPLICATION OF HUYGBNS'S PRINCIPLE. 169 

in the time dt, i. e. tlie infinitely small surface that is reached at 
the instant t -[■ dt by a disturbance starting from P at the time t. 
The envelop of aU these elementary waves wiU be the new position 
of the wave-front, and by continually repeating this construction we 
can foUow the wave in its propagation step by step. 

At the same time, the course of the rays of light becomes 
known. The line drawn from the centre of vibration P of an 
elementary wave to the point P' where it is touched by the envelop 0', 
is an element of a ray, and every new step in the construction will 
give us a new element of it. 

The physical meaning of the lines so determined need scarcely 
be recalled here. The rays serve to indicate the manner in which 
beams of light can be laterally limited. If, for example, the light 
is made to pass through an opening in an opaque screen, the 
disturbance of the equilibrium behind the screen is confined to the 
part of space that can be reached by rays of light drawn through 
the points of the opening. It must be kept in mind, however, that 
this is true only if we neglect the efi'ects of diffraction, as we may 
do when the dimensions of the opening are very large in comparison 
with the wave-length. 

If we want to lay stress on the fact that, in the above con- 
struction, we had in view the relative motion of light with respect 
to the axes of coordinates or with respect to some system to which 
these are fixed, we can speak of the relative rays of light. 

As to the elementary waves, on whose dimensions and form aU 
is made to depend, these are determined in every case by the optical 
properties and the state of motion of the medium. 

147. We are now prepared for examining the two theories of 
the aberration of light that have been proposed by Fresnel and 
Stokes. In doing so we shall confine ourselves to the annual aber- 
ration, so that the rotation of our planet around its axis wiU be left 
out of consideration. In order further to simplify the problem, we 
shall replace the motion of the earth in its annual course by a uni- 
form translation along a straight line. 

The theory of Stokes') rests on the assumption that the ether 
surrounding the earth is set in motion by the translation of this body, 
and that, at every point of the surface of the globe, there is perfect 
equality of the velocities of the earth and the ether. According 
to this latter hypothesis, the instruments of an observatory are at 
rest relatively to the surrounding ether. It is clear that under these 



1) G. G. Stokes, On the aben-ation of light, Phil. Mag. (3) 27 (1845), p. 9; 
Mathematical and physical papers 1, p. 134. 



170 V. OPTICAL PHENOMENA IN MOVING BODIES. 

circumstances the direction in which a heavenly body is observed, 
must depend on the direction of the waves, such as it is immediately 
before the light enters our instruments. Now, on account of the 
supposed motion of the ether, this direction may differ from the 
direction of the waves at some distance from the earth; this is the 
reason why the apparent position of a star will be different from 
the real one. 

In order to determine the rotation of the waves we shall now 
apply the general method that has been sketched, using a system of 
coordinates that moves with the earth. We shall denote by g the 
velocity with which the ether moves across our diagram, a velocity 
that is at the surface of the earth, if there is no sliding, and equal 
and opposite to the velocity of the earth at a considerable distance. 
The state of motion beiag stationary, this relative velocity of the 
ether is independent of the time. We shall further neglect the in- 
fluence of the air on the j)ropagation of light, an influence that is 
known to be very feeble. 

If the ether were at rest relatively to the axes, the light-waves 
would travel with the definite velocity c; every elementary wave 
would be a sphere whose radius is cdt, and whose centre lies at the 
point P from which the radiation goes forth. For the moving ether 
this has to be modified. The elementary wave still remains a sphere 
with radius celt, because in the infinitely small space in which it is 
formed, the ether may be taken to have everywhere the same velo- 
city, but while it expands, the sphere is carried along by the motion 
of the medium, in exactly the same manner in which waves of sound 
are carried along by the wind, or water waves by the current of a 
river. The elementary wave formed around a point P (Fig. 8) wiU 
therefore have its centre, not at P, but at another point Q, namely 
at the point that is reached at the time t -\- dt }iy & particle of the 
ether which had the position P at the time t. There will be a ro- 
tation of the wave-front, if the velocity g of the ether changes from 
one point of the wave to the next. 

It win suffice for our purpose to consider so small a part of 
the wave as can be admitted into the instrument of observation. A 
part of this size can be considered as plane and the velocity of the 
ether at its different points can be regarded as a linear function of 
the coordinates. Consequently, the centres of the spheres lie in a 
plane and, since the spheres are equal, the part of the new wave- 
front (/ with which we are concerned is a plane of the same direction, 
so that the rotation of the wave is equal to the rotation of a plane 6 
that is carried along by the motion of the medium. 

Let us lay the axis OX along the normal N to the wave-front 
6, drawn in the direction of propagation. Then, the direction cosines 



STOKES'S THEORY OF ABERRATION. 171 

of the normal N' to the new wave-front are easily found ^) to be 
proportional to the expressions 

dx ' oy ' dz 

We can express this result by saying that the direction of the normal 
N' is obtained if a vector of unit length in the direction of iV is 
compounded with a vector whose components are 

-"^'dt, -^-^dt, -^-dt. (249) 

dx ' dy ' dz ^ ' 

A vector which serves in this way to determine the change of a 
direction, by being compounded with a unit vector in the original 
direction, may be termed a deviating vector. 

There is one assumption which plays a very important part in 
Stokes's theory and of which thus far no mention has been made. 
Stokes supposes the motion of the ether to be irrotational, or, in 
other terms, to have a velocity potential. In virtue of this we have 

Hx ^ 3 fly Hs ^Sg, 
dy dx ' dz dx ' 

so that we can represent the components (249) of the deviating 
vector by 

-l^dt, -i^'dt, -^^dt, 
dx ' dx ' dx ' 



and the vector itself by 



lldt. (250) 



148. The velocity W of the earth being only one ten-thousandth 
part of the speed of light, aU the terms in our formulae which 

contain the factor - — -, are very smaU. So are also the terms con- 
taining the factor — , if g is one of the velocities of matter or 

ether, and v one of the velocities of light with which we are 
concerned. We shall call terms of this kind quantities of the first 
order of magnitude, and we shall neglect in the majority of cases 
the terms of the second order, i. e. those which are proportional 



to -^ or to ^ 

c v 



If we do so, the calculation of the total rotation which the 
waves of light undergo while advancing towards the earth, and which 
is a quantity of the first order, is much simplified. We have only 
to form the sum of all the deviating vectors such as (250) which 

1) Note 65. 



172 V. OPTICAL PHENOMENA IN MOVING BODIES. 

belong to the successive elements of time; the resultant vector will 
be the total deviating vector, i. e. the vector which we must com- 
pound with a unit vector in the direction of the original normal to 
the waves, in order to get the final direction of the normal. Since 

C ' 

(250) becomes 

_lli. OP' 
c dx '"^-^ ' 

and here we may replace QP' by the element FP' = ds of the ray, 
because the ratio yyp, differs from 1 by a quantity of the order -^ , 

and the factor — 5-- is also of this order of magnitude. Finally, 

cox 

we may replace ^ by -^j because the angle between ds and the 

axis of X, which coincides with the wave-normal, is a quantity of 

the order ^- The deviating vector corresponding to the element ds 

becomes by this 

--^Ids, 
c cs ' 

an expression from which all reference to the axes of coordinates has 
disappeared, and, if the ray travels from a point A to a point B, 
we have for the total deviating vector 

B 



-|/l!^^' = |(9--9-)' 



where g^i and g^ are the relative velocities of the ether at the 
points A and B. 

Now, let the point J. be at a great distance from the earth, 
and let B lie in the immediate neighbourhood of its surface. Then, 
if there is no sliding, we have 0^ = 0, whereas g^ is equal and 
opposite to the velocity W of the earth. The deviating vector becomes 



w 
c 



and we can draw the following conclusion: 

In order to find the final direction of the wave-normal (in the 
direction of the propagation) we must draw a vector equal to the 
velocity c of light in the direction of the original normal to the 
waves at A, and compound it with a vector equal and opposite to 
the velocity of the earth. If one takes into account that the normal 
at A coincides with the real direction of the light coming from a 



STOKES'S THEORY OF ABERRATION. 173 

star, it is clear tliat our result agrees with the ordinary explanation 
of aberration that is given in text-books of astronomy and that has 
been verified by observations. 

149. Unfortunately, there is a very serious difficulty about this 
theory of Stokes: two assumptions which we have been obliged to 
make, namely that the motion of the ether is irrotational and that there 
is no sliding over the surface of the earth, can hardly be reconciled. 
It is wholly impossible to do so, if the ether is regarded as incom- 
pressible. Indeed, a well known hydrodynamical theorem teacTies us 
that, when a sphere immersed in a boundless incompressible medium has 
a given translation, the motion of the medium wiU be completely 
determined if it is required that there shall be a velocity potential, 
and that, at every point of the surface, the velocity of the medium 
and that of the sphere shall have equal components in the direction 
of the normal. In the only state of motion which satisfies these 
two conditions there is a considerable sliding at the surface, the 
maximum value of the relative velocity being even one and a half 
times the velocity of translation of the sphere.^) This shows that an 
irrotational motion of the medium without sliding can never be 
realized if the medium is incompressible, and that we should have 
at once to dismiss Stokes's theory if we could be sure of the in- 
compressibility of the ether. 

The preceding reasoning fails however, if we admit the possibi- 
lity of changes in the density of the ether, and Planck has observed^) 
that the two hypotheses of Stokes's theory no longer contradict 
each other, if one supposes the ether to be condensed around celestial 
bodies, as it would be if it were subjected to gravitation and had 
more or less the properties of a gas. We cannot wholly avoid a 
sliding at the surface, but we can make it as small as we please by 
supposing a sufficient degree of condensation. If we do not shrink 
from admitting an accumulation of the ether around the earth to a 
density f" times as great as the density in celestial space, we can 
imagine a state of things in which the maximum velocity of sliding 
is no more than one half percent of the velocity of the earth, and 
this would certainly be amply sufficient for an explanation of the 
aberration within the limits of experimental errors.^) 

In this department of physics, in which we can make no pro- 
gress without some hypothesis that looks somewhat startling at first 



1) Note 66. 

2) See Lorentz, Stokes's theory of aberration in the supposition of a 
variable density of the aether, Amsterdam Proceedings ]K9a— 1899, p. 44.3 (Ab- 
handlungeu fiber theoretische Physik I, p. 454). 

3) Note 67. 



] 74 V. OPTICAL PHENOMENA IN MOVING BODIES. 

sight, we must be careful not rasUy to reject a new idea, and in 
making his suggestion Planck has certainly done a good thing. Yet 
I dare say that this assumption of an enormously condensed ether, 
combined, as it must be, with the hypothesis that the velocity of 
light is not in the least altered by it, is not very satisfactory. I am 
sure, Planck himself is inclined to prefer the unchangeable and 
immovable ether of Fresnel, if it can be shown that this conception 
can lead us to an understanding of the phenomena that have been 
observed. 

150. The theory of Fresnel, the main principle of which has 
already been incorporated in the theory I have set forth in the pre- 
ceding chapters, dates as far back as 1818. It was formulated for the 
first time in a letter to Arago^), in which it is expressly stated that 
we must imagine the ether not to receive the least part of the motion 
of the earth. To this Fresnel adds a most important hypothesis 
concerning the propagation of light in moving transparent ponderable 
matter. 

I believe every one wiH be ready to admit that an optical phe- 
nomenon which can take place in a system that is at rest, can go 
on in exactly the same way after a uniform motion of translation 
has been imparted to this system, provided only that this translation 
be given to all that belongs to the system. If, therefore, aU that is 
contained in a column of water or in a piece of glass shares a trans- 
latory motion which we communicate to these substances, the propa- 
gation of light in their interior wiU. always go on in the same 
manner, whether there be a translation or not. The case wiE. however 
be different, if the glass or the water contains something which we 
cannot set in motion. 

Now, as I said, Fresnel supposed the ether not to follow 
the motion of the earth. The only way in which this can be under- 
stood, is to conceive the earth as impregnated throughout its bulk 
with ether and as perfectly permeable to it. When we have gone 
so far as to attribute this property to a body of the size of our 
planet, we must certainly likewise ascribe it. to much smaller bodies, 
and we must expect that, if water flows through a tube, there 
is no current of ether, and that therefore, since a beam of light is 
propagated partly by the water and partly by the ether, the light 
waves, being held back as it were by the ether, will not acquire the 
full velocity of the water current. According to Fresnel's hypo- 
thesis, the velocity of the rays relative to the walls of the tube 



1) Lettre de Fresnel a Arago, Sur rinfluence du mouvement terrestre 
dans quelques phenomenes d'optique, Ann. de chim. et do phys. >) (1818), p. 57 
(CEuvres completes de Fresnel 2, p. 627). 



FRESNEL'S COEFFICIENT. 175 

(or, what amounts to the same thing, relative to the ether) is 
found by compounding the velocity with which the propagation would 
take place in standing water, with only a certain part of the 

velocity of the flow, this part being determined by the fraction 1 ^ , 

where /t is the index of refraction of the water when at rest. The 

same coefficient 1 j is applied by him to all other isotropic 

transparent substances. If ft is little different from 1, as it is in 
gases, the coefficient is very small; light waves are scarcely dragged 
along by a current of air, because in air the propagation takes place 
almost exclusively in the ether it contains. If Fresnel's coeffi- 
cient is to be nearly 1, i. e. if the light waves are to acquire almost 
the full velocity of the ponderable matter, we must use a highly re- 
fracting body. 

151. I must add two remarks. In the first place, instead of 
the propagation relative to the ether, we can as well consider that 
relative to the ponderable matter. If water which is flowing through 
a tube towards the right-hand side with a velocity tv, is traversed 
by a beam of light going in the same direction, the velocity of 
propagation relative to the ether is 



^+(^-7^)^' 



where v means the velocity of light in standing water. The rela- 
tive velocity of the light with respect to the water is got from this 
by subtracting iv, so that it is given by 

v-^,w. (251) 

It may be considered as compounded of the velocity v and a part, 
determined by the fraction — „, of the velocity with which the ether 

moves relatively to the ponderable matter, and which in our example 
is directed towards the left. 

In the second place, the above statement of Fresnel's hypo- 
thesis requires to be completed for the case of media in which the 
velocity of light depends on the frequency. When a body is in 
motion, we must distinguish between the frequency of the vibrations 
at a fixed point of the ether and the frequency with which the 
electromagnetic state alternates at a point moving with the ponderable 
matter. If, using axes of coordinates fixed with respect to the ether, 
we represent the disturbances by means of formulae containing an 
expression of the form 

aosn(t-^+p), (252) 



176 V. OPTICA!; PHENOMENA IN MOVING BODIES. 

n is tlie first of these frequencies, which may be termed the true or 
absolute one. We can pass to the other, the relative frequency, by 
introducing into this expression the coordinate with respect to an 
origin moving with the ponderable matter. If this coordinate is 
denoted by x, and if the motion of the matter takes place in the 
direction of OX with the velocity w, we have 

X = X -{■ wt, 
so that (252) becomes 

cos n\t 1 V P)- 

The coefficient of t in this expression, 



nl 



(1-3 



is the relative frequency; that it differs from n agrees with Doppler's 
principle. 

Fresnel's hypothesis may now be expressed more exactly as 
follows. If we want to know the velocity of propagation of light 
in moving ponderable matter, we must fix our attention on the 
relative frequency n of the vibrations, and we must understand by v 
and (I in the expression (251) the values relating to light travelling 
in the body without a translation, and vibrating with a frequency 
equal to n'. 

152. I have now to show that Fresnel's theory can account 
for the phenomena that have been observed. These may be briefly 
summarized as follows. First there is the aberration of light of which 
I have already spoken. Further it has been found that an astro- 
nomer, after having determined the apparent direction of a star's rays 
and their apparent frequency, can predict from these, by the ordinary 
laws of optics, and without attending any more to the motion of the 
earth, the result of aU experiments on reflexion, refraction, diffraction 
and interference that can be made with these rays. Finally, all 
optical phenomena which are produced by using a terrestrial source 
of light are absolutely independent of the earth's motion. If, by a 
common rotation of the apparatus, the source of light included, we 
alter the direction of the rays with respect to that of the earth's 
translation, not the least change is ever observed. 

It must be noticed that aU this could be accounted for at a 
stroke and without any mathematical formula by Stokes's theory, 
if only we could reconcile with each other its two fundamental 
assumptions. In applying Fresnel's views, we need some calculations, 
but these will lead us to a very satisfactory explanation of all that 
has been mentioned, with the restriction however that we must 
confine ourselves to the effects of the first order. 



VELOCITY OF A RAY OF LIGHT IN A MOVING MEDIUM. 177 

153. We shall again begin by considering the propagation of 
the wave-front, this time in the interior of a ponderable transparent 
body, whose properties may change from point to point, but which 
we shall suppose to be everywhere isotropic. For a given frequency, 
the velocity of light in the body while it is at rest will have at 
every point a definite value v, connected with the index of refrac- 
tion ji by the relation 

c 
ti = — 

^ V 

As before, we shall use axes of coordinates that are fixed to the 
earth; if we represent the progress of the waves in a diagram, this 
will likewise be supposed to move with the earth, so that the ether 
must be understood to flow across it, with a velocity which will again 
be denoted by g, but which now has the same direction and magni- 
tude at all points, being everywhere equal and opposite to the velocity 
of the earth. 

Let, as before, G be the position of a wave-front (see Fig. 8, p. 168) 
at the time t, 6' the position at the time t -\- dt, the latter surface 
being the envelop of aU. the elementary waves that have been formed 
during the time dt. If the ether were at rest in our diagram, each 
elementary wave would be a sphere having a radius vdt, and whose 
geometric centre coincides with the centre of vibration. In reality, 
according to what has been said about Fresnel's hypothesis, the 
geometric centre of the sphere, whose radius is stiU. vdt, is dis- 
placed from the centre of vibration over a certain distance, the dis- 
placement being given by the vector — | g dt. 

Let us consider the infinitely small triangle having its angles at 
the point P of the wave-front U, which is the centre of disturbance 
for the elementary wave, the point Q which is its geometric centre, 
and the point P' where it is touched by the new wave-front 6. As 

has just been said, the side P $ as a vector is given by -j g dt. The 

side QF', being a radius of the sphere, is normal to 0', and, in the 

limit, to 6. Its length is vdt. As to the side PP', this is an ele- 

pjj' 

ment of a relative ray. According to general usage, we shall call -^ 
the velocity of the ray, so that, if this is denoted by v, we have 

FT'=v'dt. 

It appears from this that, if the angle between the relative ray and 
the velocity g is represented by ■9', 



V^ = v'^-2'^lv' <iOB» + -L', 



Lorentz, Theory of electrons. 1" 



178 V. OPTICAL PHENOMENA IN MOVING BODIES. 

from which one finds, omitting quantities of the third order, i. e. of 

I a P 
the order ^, 

«' = ^, + Mcos#- J^.sin^*. (253) 

We shall have especially to consider the inverse of this quantity. 
To the same degree of approximation, it is given by 

1 Mi l9 



There is further a simple rule by means of which we can pass from 
the direction of the wave-normal to that of the relative ray and con- 
versely. The vector I'T' is the sum of the vectors TQ and QV . 
Hence, dividing the three by dt, we have the following proposition: 
If a vector having the direction of the normal to the wave and the 

magnitude v, is compounded with a vector -^, the resultant vector 

win be in the direction of the relative ray. And, conversely, if a 
vector in the direction of the ray and having the magnitude v , is 

compounded with a vector — -^, we shall find the direction of the 

normal to the wave. 

In order fuUy to understand the meaning of these propositions, 
one must keep in mind that, at every point of the medium, the re- 
lative ray and the wave can have all possible directions. The above 
results apply to aU cases. 

154. These preliminaries enable us to prove the beautiful 
theorem that, if quantities of the second order are neglected, the 
course of the relative rays is not affected by the motion of the 
earth. We have seen in what manner Huygens's principle, while 
determining the successive positions ff, e' , u", ... of a wave-front, 
also gives us the succeeding elements PP', F' F'\ F"P"', ... of a 
relative ray. If the centre of vibration of an elementary wave and 
the point where it is touched by the envelop are called conjugate 
points, we may say that a ray passes through a series of conjugate 
points succeeding each other at infinitely small distances. Now, be- 
tween any two consecutive positions of the wave-front, we can draw 
a large number of infinitely small straight lines, some joining con- 
jugate points and others not, and for each of these lines ds we can 
calculate the value of 

?^ (255) 

taking for v' the value belonging to an element of a ray having the 
direction of ds. It is easily seen that this expression (255) has one 



COURSE OP A RELATIVE RAY. 179 

and the same value for aU. lines joining conjugate points, and a greater 
value for aU other lines. Indeed, by the definition of v, the value 
is for the first lines equal to the time dt in which the light advances 
from the first position of the wave-front to the second. As to a 
line ds which is drawn between a point P of the first wave-front 
and a point Q of the second, not conjugate with P, its end Q lies 
outside the elementary wave that is formed around P, because the new 
wave-front is less curved than the elementary wave and must lie 
outside it with the exception of the point of contact. Therefore, for 
the line P$, the expression (255) must exceed the value it would 
have if Q lay on the surface of the elementary wave. 

Now, let A and H be two points of a relative ray s, at a finite 
distance from each other, and let s be any other line joining these 
points. If between A and P we construct a series of wave-fronts at 
infinitely small distances from each other, the line s is divided into 
elements each of which joins two conjugate points, whereas the ele- 
ments of s cannot be all of this kind. From this we can infer that 
the integral 

pi (256) 

taken for s will have a smaller value than the corresponding integral 
for the line &'. Thus, the course of the relative ray between two 
given points A and P is seen to be determined by the property that 
the integral (256) is smaller for it than for any other line between 
the same points. 

Substituting in the integral the value (254) we find, if we 
neglect terms of the second order, 

B B B 

J'lf = J'L^ _ jLsl^rfs. (257) 

AAA 

Here, since [iv = c, we may replace the last term by 

B 

j,fQ,ds=^{AB%\Q\, 



A 



if we understand by (AS)^^ the projection of the path AJB on the 
direction of the velocity g, a projection that is entirely determined 
by the position of the extreme points A and P. The last term in 
(257) is therefore the same for all paths leading from A to P, and 
the condition for the minimum simply requires that the first term 



A 

12' 



180 V. OPTICAL PHENOMENA IN MOVING BODIES. 

be a minimum. This term, however, contains nothing that depends 
on the velocity g; hence, the course of the ray, for which it is a 
minimum, is likewise independent of that velocity, by which our 
proposition is proved. 

In the proof we have made no assumption concerning the way 
in which v and ^i change from point to point. It applies to any 
distribution of isotropic transparent matter, and even to limiting cases 
in which there is a sudden change of properties at a certain surface. 
Consequently, for the relative rays, the law of refraction remains the 
same as it would be if the bodies were at rest (in which case the 
word „relative" might as weU be dropped). I must add that this 
proposition can easily be proved by itself, by directly applying 
Huygens's principle to the refraction at a surface, and that the 
reflexion of rays can be treated in the same manner and with the 
same result. 

155. In order to account for the phenomenon of aberration, 
one has only to combine the above results. Let P be a distant point, 
which we imagine to be rigidly connected with the earth, and to lie just 
outside the atmosphere in the free ether. At this point, the light 
coming from some star will have waves whose normal has a definite 
direction N, opposite to the direction in which the star is really 
situated. It has also a definite relative frequency, which in general 
difiers from the true or absolute one according to Doppler's principle. 

At the point P we have v == c, /x = 1. Hence, if we want to 
find the direction of the relative ray s at this place, we must com- 
pound a vector c in the direction of the wave • normal N with a 
vector g, which represents the velocity of the ether relative to the 
earth, and which is therefore equal and opposite to the velocity of 
the earth itself This construction evidently leads to a direction of 
the relative ray identical with the apparent direction of the rays as 
determined in the elementary theory of aberration. We shall there- 
fore have explained this latter phenomenon if we can show that the 
result of observations made at the surface of our planet is such that 
an astronomer (who does not think of the earth's motion), reckoning 
so far as necessary with the frequency n which shows itself to 
him, would conclude from them that the rays reach the atmosphere 
in the direction s. This is really so, because, as we have seen, the 
progress of the relative rays from P onward is exactly what would 
be the progress of the absolute rays if the earth did not move and 
the true frequency were equal to n. 

We may mention in particular that, if, in this latter case, the 
path of a ray were mapped out by means of suitably arranged screens 
with small openings, a ray can stiU pass through these openings, if 



EXPERIMENTS WITH TERRESTRIAL SOURCES OF LIGHT. 181 

the screens move with the earth. Further that if, on the immovable 
earth, the absolute rays were brought to a focus in a telescope, the 
relative rays wiU likewise converge towards this point, producing in 
it a real concentration of light. The truth of this is at once seen 
if, by means of the theorem of § 153, we determine the shape of 
the wave-fronts in the neighboui-hood of the focus. It is found that 
the convergence of the relative rays towards a point necessarily 
implies a contraction of the waves around this point. ^) 

The explanation of the fact that all optical phenomena which 
are produced by means of terrestrial sources of light are uninfluenced 
by the earth's motion, is so simple that few words are needed for it. 
It wiE suffice to observe that in experiments on interference the 
differences of phase remain unaltered. This foUows at once from our 
formula (257) for the time in which a relative ray travels over a 
certain path. If two relative rays, starting from a point A, come 
together at a point B, the lengths of time required by them are 
given by the expressions 



/ 



4?' + ?(^^), 9; 



and 

B 



/^ + i(^-S)Jol, 



where the integrals relate to the two paths. Since the last terms 
are identical, we find for the difference between the two times 






This being independent of the motion of the earth, the result of the 
interference must be so likewise, a conclusion that may be extended 
to all optical phenomena, because, in the light of Huygens's prin- 
ciple, we may regard them aU as cases of interference. 

It should be noticed, however, that the position of the bright 
and the dark interference bands is determined by the differences of 
phase expressed in times of vibration, so that the above conclusions 
are legitimate only if the motion of the earth does not affect the 
periods themselves in which the particles in the source of light are 
vibrating. This condition wiU be fulfilled if neither the elastic forces 
acting on them, nor their masses are changed. Then, in all experi- 
ments performed on the moving earth, the relative frequency at any 



1) Note 68. 



182 V. OPTICAL PHENOMENA IN MOVING BODIES. 

point of our apparatus will be equal to the frequency that would 
exist if we could experiment in the same manner on a planet having 
no translation. 

156. Presnel's coefficient 1 2, the importance of which we 

have now learned to understand, can be deduced from the theory that 
in a beam of light in a ponderable body there is an oscillatory 
motion of electric charges. Unfortunately, if these latter are supposed 
to be concentrated in separate electrons, the deduction suffers from 
the difficulties that are inherent in most molecular theories, and the 
true cause of the partial convection of light-waves by matter in 
motion does not become clearly apparent. For this reason I shall 
first consider an ideal case, namely that of a body in which the charges 
are continuously distributed. In this preliminary treatment I shall 
make light of the difficulty that we are now obliged to imagine four 
different things, thoroughly penetrating each other, so that they can 
exist in the same space, viz. 1. the ether, 2. the positive and the 
negative electricity and 3. the ponderable matter. 

For the sake of simplicity, I shall suppose that only one of the 
two electricities can be shifted from its position of equilibrium in 
the ponderable body, the other being rigidly fixed to this latter, and 
having no other motion than the common translation of the entire 
system. I shall denote by q the volume-density of the movable, and 
by q' that of the fixed charge. The body as a whole being uncharged, 
we shall have in the state of equilibrium 

q + q'=0, (258) 

and this wiU. remain true while the one charge is vibrating, unless 
it be condensed or rarefied by doing so. 

The question as to whether it be the positive or the negative 
electricity that can be displaced in the body may be left open in 
this theory. 

157. We shall suppose the movable charge to have a certain 
mass, and to be driven back towards its position of equilibrium by 
an elastic force opposite to the displacement and proportional to it; 
let q be the displacement, — fn the elastic force, and m the mass, 
both reckoned per unit of volume. 

The equations that must be applied to the problem before us 
were already mentioned in § 11. Introducing axes of coordinates that 
have a fixed position in the ether, we have 

div d = 9 + p', (259) 

div h = 0, (260) 



DEDUCTION OF FEBSNEL'S COEFFICIENT. 183 

rotd = -yh, (261) 

roth = -^(ti + 9V + pV), (^02) 

where v and v' are the velocities of the two electricities, so that 
pV + p'v' represents the convection current. 

To these formulae we must add the equation of motion of the 
vibrating electricity. If its acceleration is denoted by j, we have 

wj = -/-q + pd + |p[v-h]. (203j 

158. Let us first briefly examine the propagation of electric 
vibrations in the body when kept at rest. We may limit ourselves 
to the case that there is a displacement q^ of the movable charge in 
the direction of Y, combined with a dielectric displacement d of 
the same direction in the ether, and a magnetic force h^ parallel to 
OZ, aU these quantities being functions of x and t only. As the 
relation (258) is not violated, the equations (259) and (260) are ful- 
filled by these assumptions, and (261) and (262) reduce to 

Finally the equation of motion becomes 






m^^ = -f\ + Qd,^. (266) 



A solution of these equations is obtained by putting 

from which we find, by means of (264) and (266), 

Substituting these values in (265), we find the following formula for 
the determination of the velocity of propagation v: 



6- 



v' f — mn' 



+ 1. (268) 



159* When the body has a uniform translation with the velo- 
city w in the direction of OX, we can stiU satisfy the equations by 
suitable values of d,^, h^, q^,, but some alterations are necessary. The 
first of these relates to the convection current. Its component in 



184 V. OPTICAL PHENOMENA IN MOVING BODIES. 

the direction of OX remains 0, since both the positive and the 
negative electricity are carried along with the translation of the body, 
but, if we continue to use axes of coordinates fixed in the ether, the 
convection current parallel to T can no longer be represented 

by Q -7^ ■ The right expression for it is found as follows. If a de- 
finite point of the vibrating charge has the coordinate x at the time t, 
its coordinate at the instant t -{- dt wiU he x + w dt, so that the in- 
crement of its displacement q^ is given by 

and its velocity in the direction of F by 

d 
for which we may also write 



dt +^8x 



\3t)' 

if we use the brackets for indicating the differential coefficient for a 
point moving with the body. The convection current may therefore 
be represented by 



'(t)- 



It is clear that the acceleration is 






and that, for any quantity tp which depends on the coordinates and 

the time, we may distinguish two differential coefficients -^ and (-^ j , 

just as we have done for q^. The first is the partial derivative 
when qp is considered as a function of t and the „absolute" coordi- 
nates, i. e. the coordinates with respect to axes fixed in the ether, 
and we have to use the second symbol when the time and the „re- 
lative" coordinates, i. e. the coordinates with respect to axes moving 
with the body, are taken as independent variables. The relation be- 
tween the two quantities is always expressed by the formula 



m 



As to the differential coefficients with respect to x, y, z, each of 
these has the same value, whether we understand by x, y, z the ab- 
solute or the relative coordinates. 

The second alteration which we have to make is due to the last 



DEDUCTION OF FRBSNEL'S COEFFICIENT. 185 

term in (23). On account of its velocity w in the direction of OX, 
fhe charge q wiU be acted on by a force 

— — wh,, 

parallel to OY, and this force must be added on the right-hand side 
of the equation of motion. 

In virtue of the assumptions now made, p + p' again remains 
during the vibrations, and (259) and (260) are satisfied. The equa- 
tion (264) can be left unchanged, but (265) and (266) must be re- 
placed by 



d 
and 






m 



['^)--t\ + 9^-'i9K 



The three formulae are somewhat simplified if we choose as in- 
dependent variables the time and the relative coordinates and if, at 
the same time, we put 

■ h, = d.;. 



c 



dx 

m{ 



Applying the relation (269) to d^ and h^, we find 

^ = _ i_ (^\ 
dx c \8t)' 

The first and the third of these equations have the same form as 
(264) and (266). Hence, if we put 

fi;=aoosn(t-^), (270) 

understanding by x the relative coordinate, we have, corresponding 
to (267), 

"z = Yy' ^y'^ f- mn' V ' 
by which the second equation becomes 

V' f — TOW V 

Comparing this with (268) we see that, for a definite value of the 
frequency n, we may write 

4 = 4 + 2"- 



186 v. OPTICAL PHENOMENA IN MOVING BODIES. 

As we , continually neglect quantities of the second order, we may, 
in the last term, replace v by v. By this we get 



c 


e w 




v 


" V "f" T ' 






tov^ 


w 


V — 


—2- = V — 


p' 



if ft is the index of refraction for the stationary body. 

It must be kept in mind that in (270) x means the relative 
coordinate. Therefore, n is the relative frequency, and v' the speed 
of propagation relative to the ponderable matter. The velocity of 
light with respect to the ether is 



v' +tv = v + (l — -ijw, 



in accordance with Fresnel's hypothesis. 

160. I have now to show you in what manner the same result 
may be derived from the theory of electrons. For this purpose we 
might repeat for a moving system all that has been said in Chap. IV 
about the propagation of light in a system of molecules. We shall 
however sooner reach our aim by following another course, consisting 
in a comparison of the phenomena in a moving system with those 
that can occur in the same system when at rest. 

In this comparison we shaU avail ourselves of the assumptions 
that have been made in Chap. IV. 

In the absence of the translatory motion, the problem may be 
stated as follows. In the molecules of the body there are electric 
moments p changing from one molecule to the next, and variable 
with the time. On account of its moment, each molecule is surrounded 
by an electromagnetic field, which is determined (§ 42) by the 
potentials 

in \ dx r dy 

[p] 
incr 



q, = i ^ -- + A M . A W. 1 

^ i.^ \ Ti-v ar dy r dz r \' 

a = 



X, y, 2 being the coordinates of the point considered, r its distance 
from the molecule, and the square brackets reminding us that we 
have to do with retarded potentials. The electric force d and the 
magnetic force h are given by the following formulae, to be deduced 
from (33) and (34), 

d L_rnl 4- J- crrad \~ ^?J 4- ^ ^^'^^ -I- ^ fPJ I h^lW 



LOCAL TIME. 187 



After having compounded with each other the fields produced by all 
the molecules of the body, we must add one field more, namely that 
which is due to external causes, and which I shall represent by d^, h^. 
It satisfies the equations 



div dj, = 0, 
div h(, = 0, 

™t ho = - tip, 

rot do = — — hn 



c 



(273) 



Lastly, we have to consider the equations of motion of the electrons 
which, by their displacement, bring about the electric moments p. 
Let each molecule contain a single movable electron e, whose dis- 
placement q gives rise to an electric moment 

p = eq. (274) 

If the symbol S relates to the superposition of the fields of aR the 
surrounding particles, and if — f{\ is the elastic force, — gf q a re- 
sistance to the motion, the equation of motion is 

m'i^ = eSA-\- eA^-ff\ — g'i\. (275) 

161. In the theory of the system moving with a velocity W we 
may avail ourselves with great advantage of the transformation that 
has already been used in § 44. 

Taking as independent variables the coordinates x', y, z with 
respect to axes moving with the system, and the „local" time 

<'=^-^(w,^'+w,2/'+wy), (276) 

we find the equations (104) — (107) for the vectors d' and h', which 
now take the place of d and h. It is true that the new formulae 
have not quite the same form as (33) — (36), and that the term 

— grad (W • a'), which makes the difference, must not be omitted, 

I w I 
being of the first order of magnitude with respect to ■ — -. but not- 
withstanding this it is found that the field caused by an electric 
moment is determined by the formulae') 

H'_ 1 rnl -^ ^ erad I -^ "^"J -^ ^ ^"^^ 4- -- ^^-^ 



1) See Note 26. 



1 8S V. OPTICAL PHENOMENA IN MOVING BODIES. 

h' = 7^ rot ( - [P] I ; 

exactly corresponding to (271) and (272). 

It is scarcely necessary to repeat that the symbols grad and rot 
have the meaning that has been specified in § 44, and that, if we 
want to calculate d' and h' for a point {x', y , /) at a distance r 
from the polarized particle, and for the instant at which the local 
time of this point has a definite value t' , we must take for p, p, p 
the values existing at the moment when the local time of the particle 

IS t 

c 

The field produced by causes outside the body is again subjected 
to the fundamental equations for the free ether. Expressed in terms 
of our new variables, these are 



div d; 


= 0, 




div \ 


= 0, 




rot hj' 


1 

c 


K, 


rot d„' 


= — 


\:^. 



0? 

as is found by making p = in (100) — (103). The form of these 
equations is identical with that of (273). 

The equation of motion of an electron must now contain the 

electromagnetic force --[w-h], which is due to the translation W, so 

that we must write for the total force acting on unit charge 

d + |[w.h]. 

This, however, is precisely the vector which we have called d'. 
Consequently, if we suppose that the elastic force determined by the 
coefficient /", and the resistance measured by y, are not modified by 
the translation, we may write for the equation of motion 

TOCJ = e2;d'+edo'-/'q-5fq, 

where the sign S has the same meaning as in (275). 

It should be noticed that the relation (274) remains true, and 
that at a definite point of the moving system, the differential coeffi- 
cients with respect to t are equal to those with respect to i . On 
account of this we may attach to the dots in the above equation the 
meaning of partial differentiations with respect to i . They must be 
understood in the same sense in the preceding formulae. 



THEOREM OF CORRESPUNDINa STATES. 189 

162. It appears from what has been said that, by the intro- 
duction of the new variables, all the equations of the problem have 
again taken the form which they have when there is no translation. 
This at once leads to the following conclusion: 

If, in the system at rest, there can exist a state of things in 
which d, h and p are certain functions of x, y, z and t, the moving 
system can be the seat of phenomena in which the vectors d', h', p 
are the same functions of the relative coordinates x, y , s' and the 
local time t' . 

The theorem may be extended to the mean values of d, h or 
d', h', the electric moment P per unit of volume, and also to the 
vector D which we have introduced in § 114, compared with a similar 
one that may be defined for the moving system. If, for the one 
system, we put 

d = E, h = H, D = E + P, 
and for the other 

d'=E', fr=H', D'=E'+P, 

the result is, that for each state in which E, H, D are certain func- 
tions of X, y, z, t, there is a corresponding state in the moving system, 
characterized by values of E', H', D' which depend in the same way 
on x', y, z', t'. 

163. The value of Fresnel's coefficient foUows as an immediate 
consequence from this general theorem. Let us suppose that in a 
transparent ponderable body without translation, there is a propagation 
of light waves, in which the components of E and H are represented 
by expressions of the form 

a cos nit ^ + P), 

where «, /3, y are the direction cosines of the normal to the wave, 
and V the velocity of propagation. Then, corresponding to this, we 
may have in the same body while in motion phenomena that may 
likewise be described as a propagation of light waves, and which 
are represented by expressions of the form 

r, ax + &y' ■\-ye . \ 
a cos n [t ^^^ 1- jpj, 

i. e., on account of (276), 

^ eos^(^- ^"'^^^y'i^^^' - ''"' + y + ^^' + V). 

If we put here 

_^ 1 J*k - iL 1 4- *!' = il J^ I *= = _L. (277) 



190 V. OPTICAL PHENOMENA IN MOVING BODIES, 

witt the condition 



the formula becomes 

a cos n 



«'2+^'2 + /2=l, (278) 

showing in the first place that v' is the velocity of propagation re- 
lative to the moving body, and in the second place that n is the 
frequency at a point moving with it. Hence, if we take v and v for 
the same value of n, we are sure to compare the speed of propa- 
gation in the two cases for equal relative frequencies. 

Neglecting the square of W, we easily find from (277) and (278) 

J_ _ _L 4_ 9 "W^ + pwy +yw, _ _!_ _L 2 ^ 

where W„ is the component of the velocity of translation along the 
wave-normal. It may be observed that, since «', j3', y' differ from a, 

ji, y only by quantities of the order , we may take the normal 

such as it is in the moving system. 
Further: 

i_ = Jl 4- ^ 

v^ w 

so that we have been led back to our former result. 

164. The hypothesis advanced by Fresnel has been confirmed 
by Fizeau's observations on the propagation of light in flowing 
water'), and, still more conclusively, by the elaborate researches of 
Michelson and Morley on the same subject.^) In these experiments 
the water was made to flow in opposite directions through two 
parallel tubes placed side by side and closed at both ends by glass 
plates; the two interfering beams of light were passed through these 
tubes in such a manner that, throughout their course, one went 
with the water, and tbe other against it. 

In order to calculate the change in the differences of phase caused 
by the motion of the fluid, it is necessary to know the velocity 
of propagation of the light relative to the fixed parts of the appa- 

1) H. Fizeau, Sur les hypotheses relatives a I'lSther lumineux, at sur una 
exp&ience qui parait demontrer que le mouvement des corps change la vitease 
aveo laquelle la lumiere se propage dans leur int&ieur, Comptes rendus 33 
(1851), p. 349; Ann. d. Phys. u. Chem., Erg. 3 (1853), p. 457. 

2) A. A. Michelson and E. W. Morley, Influence of motion of the me- 
dium on the velocity of light, Amer. Journ. of Science (3) 31 (1886), p. 377. 



APPLICATIONS OF THE THEOREM OF CORRESPONDING STATES. 191 

ratus.i) If T is the period of vibration of tte source of light, the 
preceding theory gives the following expression for the velocity in 
question 

-^^^V-J^ + jT/t- 

Here the velocity of the flow of water is represented by w, and we 
must take the upper or the under signs, according as the light goes 
with or against the stream. I must add that the last term, which 
depends on the dispersive power of the fluid, has been omitted by 
Michelson and Morley in the comparison of their experiments with 
the theory. If it is taken into account, the agreement becomes some- 
what worse; it remains however fairly satisfactory, since the influence 
of the term is but small. 

165. After having found Fresnel's coefficient, we may apply 
it to various phenomena, as has already been shown in §§ 152 — 155. 
The discussion of many a question may, however, also be based 
directly on the theorem of corresponding states without the inter- 
vention of the coefficient. 

If, for instance, the state of things in the system that is kept 
at rest, is such that in some parts of space both the electric and the 
magnetic force are continually zero, the corresponding state in the 
moving system will be characterized by the absence, in the same 
regions, of d' and h', and this involves the absence of d and h. There- 
fore, the geometrical distribution of light and darkness must be the 
same in the two systems, always provided that the comparison be 
made for equal relative frequencies. 

An interesting example is afforded by a cylindrical beam of 
light. The generating lines of its bounding surface, i. e. the relative 
rays, may have the same course in the two systems, even when the 
beams are reflected or refracted, so that the translation has no in- 
fluence on the laws of reflexion and refraction for the relative rays. 
Nor can it change the position of the point where the rays are 
brought to a focus by a mirror^) or a lens, and the principle also 
shows that the place of the dark fringes in experiments on interference 
must remain unaltered. 

The condition that is necessary for these conclusions, namely that 
the relative frequencies be equal in the two cases, will always be 
fulfilled if the source of light has a fixed position with respect to 
the rest of the apparatus, sharing its translation or its immobility. 

166. It is important to notice that the foregoing results are 
by no means limited to isotropic bodies. The case of crystals may 

1) Note C9. 2) Note 70. 



192 V. OPTICAL PHENOMENA IN MOVING BODIES. 

easily be included by conceiving either some appropriate regular 
arrangement of the particles, or a want of isotropy in the structure 
of the individual molecules, revealing itself in the elastic forces being 
unequal for different directions of the displacement of an electron. 
The latter assumption would require us to represent the components 
of the elastic force by expressions of the form 

~{fn^^ + fl2^y+flB^^), 
-if21^a:+f22% + f2s'\^), 

with f2i = fi2j fi2 = fiiy /is^/si) ^^^ foi' the proof of the theorem of 
corresponding states it would be necessary to consider the coeffi- 
cients f as unaffected by a translation of the system. 

After having shown that, in the phenomena of double refraction, 
the course of the relative rays is not altered by the motion of the 
earth, one can also examine what becomes of Fresnel's coefficient 
in the case of crystalline bodies. The result may be expressed as 
follows: 

If, foi a definite direction s of the relative ray, u and u are the 
velocities of this ray in a crystal that is kept at rest and in the 
same body when moving, then 

, v,^ 

where W^ is the component of the velocity of translation in the 
direction of the ray.^) 

167. Thus far we have constantly neglected terms of the second 

order with respect to '■ — , and in fact in nearly aU the experiments that 

have been made in the hope of discovering an influence of the earth's 
motion on optical phenomena, it would have been impossible to 

detect effects proportional to —^- There are, however, some exceptions, 

and these are of great importance, because they give rise to difficult 
and delicate problems, of which one has not, as yet, been able to 
give an entirely satisfactory solution. 

We have in the first place to speak of a celebrated experiment 
made by Michelson^) in 1881, and repeated by him on a larger scale 
with the cooperation of Morley*) in 1887. It was a very bold one. 



1) Note 71. 

2) A. A. Miohelson, The relative motion of the earth and the lumini- 
ferous ether, Amer. Journ. of Science (3) 22 (1881), p. 20. 

3) A. A. Michelson and E. W. Morley, Amer. Journ. of Science (3) 34 
(1887), p. 333. 



MICHELSON'S EXPERIMENT. 



193 



B 



two rays of light having been made to interfere after having travelled 
over paths of considerable length in directions at right angles to 
each other. Fig. 9 shows the general arrangement of the apparatus. 
The rays of light coming from the 
source L are divided by the glass 
plate P, which is placed at an angle 
of 45", into a transmitted part FA 
and a reflected one PjB. After ha- 
ving been reflected by the mirrors 
A and B, these beams return to 
the plate P, and now the part of 
the first that is reflected and the 
transmitted part of the second pro- 
duce by their interference a system 
of bright and dark fringes that is 
observed in a telescope placed on 
the line PC. 

The fundamental idea of the experiment is, that, if the ether 
remains at rest, a translation given to the apparatus must of ne- 
cessity produce a change in the diflPereuces of phase, though one 
of the second order. Thus, if the translation takes place in the 
direction of FA or AP, and if the length of PA is denoted by L, 

a ray of light will take a time ^^p^ for travelling along this path 



ff 



c 

Fig. 9. 



C+|W 



in one direction, and a time 
The total time is 



c — w 



for going in the inverse direction. 



2ic 



or, up to quantities of the second order, 

^(i+S). (2'») 

so that for the rays that have gone forward and back along PA 
there wiU be a retardation of phase measured by 

2£w' 



There is a similar retardation, though of smaller amount, for the other 
beam. In order to see this by an elementary reasoning, one has 
only to consider that a ray of this beam, even if it returns, as I shall 
suppose it to do, to exactly the same point of the plate P, does not 
come back to the same point of the ether, the point of the glass 
having moved, with the velocity w of the. earth's translation, over a 
certain distance, say from P to P', while the light went from P 



Lorentz, Theory of electrons. 



13 



194 V. OPTICAL PHENOMENA IN MOVING BODIES. 

to B and back. If Q is the point in the ether where the ray reaches 
the mirror B, we may say with sufficient approximation that the 
points P, Q, P' are the angles of an isoscele triangle, whose height 
is L (since the distances PA and PB in the apparatus were equal) 

and whose base is — ' -• The sum of the sides PQ and QP' is 



c 



so that we may write 



2yu+-^']f- 



2L 

e 



(1 + £) (280) 



for the time required by the second beam. 

It appears from this that the motion produces a diiference of 
phase between the two beams to the extent of 

and this may be a sensible fraction of the period of vibration, if L 
has the length of some metres. 

The same conclusion may be drawn somewhat more rigorously 
from the general formula (254). The time during which a relative 
ray travels along a certain path s is found to be 

Here the first term represents the time that would be required if 
there were no translation, and in the problem now before us the 
second has equal values for two paths beginning and ending at the 
same points, so that we have only to consider the last term, for 
which, using our present notation and putting ju. = 1, we shall write 

|jJ(l + cos^^)is. (281) 

The paths for which this integral must be calculated may be taken 
to be the straight lines indicated in Fig. 9.^) According to what has 
been said, COST'S- has the value 1 all along PAP, and the value at 
every point of PBP. Therefore our last expression really takes the 
two values given by (279) and (280). 

Now the difference of phase that is due to the motion of the 
earth must be reversed if, by a rotation of the apparatus, the path 
of the first ray is made to become perpendicular to the translation, 
and that of the second to be parallel to it. Hence, if the phenomena 

1) Note 72. 



DIMENSIONS OF A BODY CHANGED BY A TRANSLATION. 195 

follow the above theory, such a rotation must produce a change 
determined by 

-f (282) 

in the differences of phase, and a corresponding shifting of the 
interference bands. 

In the original apparatus of Michelson the length L was rather 
too small to bring out the effect that was sought for, but in the 
later experiments made with Morley the course of the rays was 
lengthened considerably. They were repeatedly thrown forwards and 
back by mirrors having suitable positions on different sides of the 
plate F, and which, together with the other parts of the apparatus, 
the source of light and the telescope included, were mounted on a 
slab of stone floating on mercury. For each of the rays the lines 
along which it had to travel successively nearly coincided, so that 
cos^^ may be regarded as constant for the entire course of a ray. 
If the values of this constant for the two beams are first 1 and 0, 
and afterwards, after a rotation of 90", and 1, the change under- 
gone by the differences of phase can be found from (281); it may 
still be represented by (282), if we understand by 2i the whole 
length of one of the rays. As this length amounted to about 
22 metres, the value of (282) is equal to 0,4 times the time of 
vibration of yeUow light, and a sensible shift of the bands could 
therefore be looked for. In no case, however, the least displacement 
of such a kind that it could be attributed to the cause above explain- 
ed was observed. A similar result was subsequently obtained by 
Morley and Miller^j, who came to the conclusion that, if there is 
any effect of the nature expected, it is less than one hundredth part 
of the computed value. 

168. In order to explain this absence of any effect of the earth's 
translation, I have ventured the hypothesis, which has also been pro- 
posed by Fitz Gerald, that the dimensions of a solid body undergo 

slight changes, of the order — j-, when it moves through the ether. 

If we assume that the lengths of two lines L^ and L^ in a ponderable 
body, the one parallel and the other perpeudicular to the translation, 
which would be equal to each other if the body were at rest, are 
to each other in the ratio 

1 = 1+5 (283) 

during the motion, the negative result of the experiments is easily 
accounted for. Indeed, these changes in length wiU produce an 

1) E. W. Morley and D. C. Miller, Report of an experiment to detect 
the Fitz Gerald-Lorentz effect, Phil. Mag. (6) !» (1905), p. 680. 

13* 



196 V- OPTICAL PHENOMENA IN MOVING BODIES. 

alteration in the phases of the interfering rays, amounting to a rela- 
tive acceleration 



for the ray that is passed along the line having the direction of the 
earth's motion, and this acceleration vrill exactly countei-balance the 
changes in phase which we have considered in the preceding para- 
graph. 

The hypothesis certainly looks rather startling at first sight, but 
we can scarcely escape from it, so long as we persist in regardiDg 
the ether as immovable. We may, I think, even go so far as to say 
that, on this assumption, Michelson's experiment jprows the changes 
of dimensions in question, and that the conclusion is no less legiti- 
mate than the inferences concerning the dilatation by heat or the 
changes of the refractive index that have been drawn in many other 
cases from the observed positions of interference bands. 

169. The idea has occurred to some physicists that, like an 
ordinary mechanical strain, the contractions or dilatations of which 
we are now speaking, might make a body doubly refracting, and 
Rayleigh and Brace have therefore attempted to detect a double 
refraction produced by the motion of the earth. Here again the 
search has been in vain; no trace of an effect of the kind has 
been found. 

With a view to this question of a double refraction, and for 
other reasons, it seems proper to enter upon a discussion of the electro- 
magnetic phenomena in a moving system, not only, as we did at 
first, for velocities very small in comparison with the speed of light 
c, but for any velocity of translation smaller than c. Though the 
formulae become somewhat more complicated, we can treat this pro- 
blem by much the same methods which we used before. 

Our aim must again be to reduce, at least as far as possible, 
the equations for a moving system to the form of the ordinary for- 
mulae that hold for a system at rest. It is found that the trans- 
formations needed for this purpose may be left indeterminate to a 
certain extent; our formulae will contain a numerical coefficient I, 
of which we shall provisionally assume only that it is a function 
of the velocity of translation tv, whose value is equal to unity 
for w = 0, and differs from 1 by an amount of the order of magnitude 

-r for small values of the ratio — 
c' c 

If X, y, s are the coordinates of a point with respect to axes 

fixed in the ether, or, as we shall say, the „absolute" coordinates, and 

if the translation takes place in the direction of OX, the coordinates 



NEW VARIABLES. 



197 



witli respect to axes moving with the system, and coinciding with 
the fixed axes at the instant t = 0, will be 

x^ = x-wt, y, = y, z^ = z. (284) 

Now, jnstead of x, y, z, we shall introduce new independent variables 
differing from these „relatiTe" coordinates by certain factors that are 
constant throughout the system. Putting 



I define the new variables by the equations 

x'=^Ux^, y'=^ly^, z'=lz^ 
or 

x'=ld{x — wt), y'=ly, z'=lz, 

and to these I add as our fourth independent variable 



*'=T^- 



■ lil-, (x — 
c 



(x — wt) = M(t 2 a;j ■ 



(285) 

(286) 
(287) 

(288) 



We shall again understand by U the velocity relative to the moving 
axes, so that the components of the absolute velocity are 

% + ^, ",- "., 
and we shall introduce a new vector u' whose components are 

<='I^'K, ^'-^^»y, "; = /*=".• (289) 

Let us put, similarly, 

9' = ^.9, (290) 

and let us define two new vectors d' and h' by the equations 

Then the ftmdamental equations take the form^) 

div d = (^1 ^,-j () , 

div' h' = 0, 

rot'h' = l(U+,'u'), 
1 gh' 



(291) 



rot' d' = 



di 



(292) 



The meaning of the symbols div', rot' and grad', the last of 
which we shall have to use further on, is similar to that which we 
formerly gave to div, rot and grad, the only difference being that 



1) Note 73. 



198 y- OPTICAL PHENOMENA IN MOVING BODIES. 

the differential coefficients with respect to x, y, z (taken for a con- 
stant P) are replaced by those with respect to x, y, z (taken for a 
constant value of ^'). ') 

As to the force f with which the ether acts on unit of electric 
charge, its components are found to be 



f. = ?^d; + ^^ • I ("/ •'Z - < ^') + ^' ? ("/ ''/ + < ^'^^ 






(293) 



The determination of the field belonging to a system of electrons 
may again be made to depend on a scalar potential <f>' and a vector 
potential a'. If these are defined by the equations 



A'a' 1 3'a' 1 ,. 



(294) 



in which the symbol A' stands for 

8^ 2^ a^ 

we shall have^) 

h' = rot' a'. (296) 

The analogy between these transformations and those which we for- 
merly used, is seen at a glance. The above formulae are changed into 
those of §§ 44 and 45 by neglecting aU terms which are of an order 

higher than the first with respect to — , by which Ic and I both take 

tlie value 1. In the present more general theory, it is the variable f 
defined by (288) that may be termed the local time. 

It is especially interesting that the final formulae (292) and 
(294) — (296) have exactly the same form as those which we deduced 
for small values of w. They differ from the equations for a system 
without translation in the manner pointed out in §§ 44 and 45, but, as 

1) In a paper „tjber das Doppler'sche Princip", published in 1887 (Gott. 
Nachrichten, p. 41) and which to my regret has escaped my notice all these 
years, Voigt has applied to equations of the form (6) (§ 3 of this book) a trans- 
formation equivalent to the formulae (287) and (28rf). The idea of the trans- 
formations used above (and in § 44) might therefore have been borrowed from 
Voigt and the proof that it does not alter the form of the equations for the 
free ether is contained in his paper. 

2) Note 74. 



MOVma ELECTROSTATIC SYSTEM. 199 

regards the form of the equations, the consideration of greater velocities 
of translation has not been attended by any new complications. 

170. The problem is greatly simplified when we consider an electro- 
static system, i. e. a system of electrons having no other motion 
than the common translation w. In this case a' = 0, and consequently 
h' = 0. The scalar potential tp', the vector d', and the electric force 
f are determined by 

A'(jp'=-9', (297) 

d' = — grad'qo', (298j 

These equations admit of a simple interpretation. Let us compare 
the moving system J^, the position of whose points is determined by 
the relative coordinates x^, y^, z^, with a system So ^^^^ ^^^ ^^ 
translation, and in which a point with the coordinates x', y, z cor- 
responds to the point (x^, y^, z^) in g, so that, as is shown by (286), 
S is changed into j^q if the dimensions parallel to the axis of x are 
multiplied by kl, and the dimensions which have the direction of y 
or that of z, by I. Then, if dS and dS' are corresponding elements 
of volume, we shall have 

dS'^JoPdS, (299) 

so that, if we suppose corresponding elements of volume to have 
equal charges, the density at a point of ^q will be given by the 
quantity q' that has been defined by (290). 

It follows that the equation which determines the scalar potential 
in gg has the same form as the equation (297) which we have found 
for qo', and that, therefore, this latter quantity has, at a point P of 
S, the same value as the ordinary scalar potential at the correspond- 
ing point Pq of So- The equation (298) further teUs us that the 
same is true of the vector d' at the point P and the dielectric dis- 
placement at the point P^. But, in order to find the components 

P 
of the electric force in Sj "^^ must multiply those of d' by P, j- , 

72 

-r , whereas, in the system ^q, the components of the electric force 

are immediately given by those of the dielectric displacement. Hence, 
there is between these electric forces a relation that is conveniently 
expressed by the formula 

the coefficients between the brackets being those by which we 
must multiply the components of the force in ^^ i^ order to get 
those of the force in g. Since corresponding elements have equal 



200 V. OPTICAL PHENOMENA IN MOVING BODIES. 

charges, the same relation exists between the forces actijig on cor- 
responding electrons. 

It is to be observed that corresponding electrons in the two 
systems occupy corresponding parts of space, and that, while their 
charges are equal, they are geometrically dissimilar; if the electrons 
in ^^ are spheres, those in §„ are lengthened ellipsoids. 

Let us also remember that the potential at a poiat F^ of }^q, 
and, consequently, the quantity (p' at the corresponding point P of 
g, can be calculated by means of the formula 

where we have denoted by / the distance between a point Q^ of the 
element dS' and the point P^. The integration is to be extended to 
aU elements in ^^ where there is a charge. 

The comparison of a moving system with a stationary one will 
be found of much use in the remaining part of this chapter, and 
it is therefore proper to settle once for all that, if we speak of 
;«> and ^p, we shall always mean two systems of this kind, and that 
the index wiU constantly serve to denote the stationary system. 

171. With a view to later developments it wiU be well to put 
the foregoing statements in yet another form. Let us, for a while, 
discard aU thoughts of the imaginary system j»5q, and confine ourselves 
to the system f-^ with which we are really concerned. We may intro- 
duce for this, as we have already done, the quantities x, y, s, and 
we may even use them for the determination of the position of a 
point, because they are related iu a definite manner to the values of 
x^, y^, s,.. Let them be called the effective coordinates, and let us 
define the effective distance between two points whose effective coordi- 
nates are x^, y^, z^, x^, y^, g^' as the quantity 



/ = Vix^ - x^y + (y,' - y,y + (,,'- z,y. 

If dx^, dy^, dg^ are infinitely small increments of the relative coor- 
dinates, the corresponding increments of the effective coordinates 
wiU be 

dx' = Iddx^, dy = Idy^, ds = Idz^, 

and, of course, the parallelepiped having dx, dy, ds for its edges 
may be said to be determined by these increments dx, dy', dz . If, 
instead of the ordinary unit of volume, we choose a unit W times 
smaller, the volume of the parallelepiped will be expressed by the 
product dx'dy'dz, and, on the same scale, an element of any form 
that is given in ordinary measure by dS, wiU have a volume 

dS' = JcPdS. (302) 



MOVING AND STATIONARY ELECTROSTATIC SYSTEM COMPARED. 201 

This is equal to the dS' in the equation (299), but the symbol has 
got a new meaning. Having already used several times the vi^ord 
„effective", I shall now — only for the sake of uniformity and without 
attaching any further meaning to the words — call dS' the effective 
element of volume. A point within dS will also be said to belong 
to the effective element dS'. 

Finally, if the charge gdS of an element dS is divided by the 
magnitude of the effective element dS', we get the quantity q that 
is defined by (290). For this reason it is not inappropriate to call q 
the effective density of the charge. 

It wiU now be clear that the operations involved in the symbol 
on the right-hand side of the equation (301) may be described in 
terms relating only to the real system, the denominator r' being the 
effective distance between a point of the effective element dS' and 
the point P for which we want to calculate (p' This potential having 
been determined, its partial differential coefficients with respect to the 
effective coordinates, taken with the signs reversed, wiU represent 
the components of the vector d'. 

It is only for moving systems that we have had reason to 
distinguish between the effective coordinates and the „true" coordi- 
nates, the effective elements of volume and the „true" ones, etc.; as 
soon as w = 0, we shall have x = x^= x, y = yr= y, ■^' = 3^.= z, 
dS'=dS, q'=q, etc. Yet, for the very reason of these equalities, 
we are free also to speak of the effective coordinates, the effective 
density, etc. in the case of a stationary system; only, we must not 
forget that in this case these quantities are identical with the true 
coordinates, the true density, etc. Similarly, we may always speak 
of the vector d', remembering that it is identical with d when there 
is no translation. 

I have dwelled at some length on these questions of denomi- 
nation, because in intricate problems a proper choice of terms is 
of much value. That which we have now made enables us to con- 
dense into few words what was said in the last paragraph about the 
systems {^ and ^^, namely: In two electrostatic systems, the one moving 
and the other not, in which the effective density of the electric 
charge is the same function of the effective coordinates, the vector d' 
will be the same at corresponding points, and the forces will be re- 
lated to each other in the way expressed by (300). 

172. Let us now return from this digression to the hypothesis 
by which we have tried to account for the result of Michelson's 
experiment. We can understand the possibility of the assumed 
change of dimensions, if we keep in mind that the form of a 
solid body depends on the forces between its molecules, and that, in 



202 V. OPTICAL PHENOMENA IN MOVING BODIES. 

all probability, tbese forces are propagated by the intervening ether 
in a way more or less resembling that in which electromagnetic 
actions are transmitted through this medium. From this point of 
view it is natural to suppose that, just like the electromagnetic forces, 
the molecular attractions and repulsions are somewhat modified by a 
translation imparted to the body, and this may very well result in a 
change of its dimensions. 

Now, it is very remarkable that we find exactly the amount of 
change that was postulated in § 168, if we extend to molecular 
actions the result found for the electric forces, i. e. if, comparing two 
systems of molecules g and gp in which the particles have the same 
effective coordinates, we admit for the molecular forces the relation 
expressed by (300). 

Indeed, this equation implies that if F(g(,) = 0, F(S) is so like- 
wise, so that when, in the system ^j, each molecule is in equilibrium 
under the actions exerted on it by its neighbours, the same will be 
true for the system [«i. Hence, taking for granted that there is but 
one position of equilibrium of the particles, we may assert that, in 
the moving system ^, the molecules wiU take of themselves the 
arrangement corresponding, in the manner specified by (286), to 
the configuration existing in g,,. Since x', y', z are the true coordi- 
nates in this latter system, and x^, y^, g^ the relative coordinates in 
|-5, the change of dimensions in different directions is given by the 
coefficients in (286), and the two lines of which we have spoken in 
§ 168 will be to each other in the ratio 

which agrees with the value (283), if quantities of an order higher 
than the second are neglected. 

173. It is a matter of interest to inquire whether our assump- 
tions demand the same change of dimensions for bodies whose shape 
and size depend in a smaller or greater measure on their molecular 
motions. As a preliminary to this question, I shall consider a 
system of points having, besides a common translation w, certain 
velocities U. For each of them the coordinates x^, y^, z^ are definite 
functions of the time t, and 

-^ = II -^ - II ^ _ II 
dt ^' dt y dt ~ "»• 

But we may also say that for each the effective coordinates x, y', s 
are functions of the local time t' , which I shall henceforth also term 
the effective time, and we may calculate the differential coefficients 
of x, y, z with respect to t' in terms of those of x^, y^, z^ with 



MOLECULAR MOTION. 203 

respect to t. In doing so I shall suppose the velocities u^, u, , u^ to 

be so small that terms that are of the order of magnitude ^ compared 

with those I am going to write down, may he neglected. Then the 
result is^) 



dx _ p dx^ dp' , dy^ 
dt' "' dt ' dt' ~ '^ ~dt ' 


dz' 1 dz^ 
dt' " dt ' 


d^x' k'd'x^ d^y' Tc^ d^y^ 
dt'^ I dt^ ' dt'^ I dt^ > 


d^/ k^ d^Zr 

dt' ^ ~ r ~df 



(303) 

The first set of equations shows that -rrr , -Mr , 4-t are the compo- 
^ dt ' dt ' dt ^ 

uents of the vector u' that has been defined in § 169, and it appears 

from the second set that, if there are two systems of points g and 

^0 moving in such a way that in both the effective coordinates are 

the same functions of the effective time, we shall have the following 

relation between the accelerations j 

J(S) = (,i, i,, ^)j(So)- (304) 

This formula, in which the mode of expression is the same as that 
which we have used in (300), follows immediately from (303), the 

components of the acceleration in g^ being yrTi, 7/2, JVi! ^^*^ those 

d X d 11 d z 
of the acceleration in g -j-f , -^ , -jrf ■ 

174. It remains to apply this to a body in which molecular 
motions are going on. At ordinary temperatures the velocities of 
these are so smaE in comparison with that of light, that the approxi- 
mations used in the above formulae seem to be allowable. On the 
same ground we may regard the interactions of the molecules to be 
independent of the velocities U, and to be determined solely by the 
relative positions and the velocity of translation w. 

Let S and ^^ be two systems of molecules moving in such a 
manner that in both the effective coordinates are the same functions 
of the effective time. Let us fix our attention on two corresponding 
particles P and P^ in the positions which they occupy for a definite 
value, say f, of the effective time. If we wish to know the simul- 
taneous positions of the neighbouring particles of go; which are suf- 
ficiently near P„ sensibly to act on it, we have only to consider the 
values of their coordinates x, y , z for the same value i_ of the effec- 
tive, i. e. in this case, the true time. It is otherwise with the moving 
system g. Here the instants for which the effective (i. e. now, the 
local) time, has a definite value T at different points, are not simul- 

1) Note 75. 



204 V. OPTICAL PHENOMENA IN MOVING BODIES. 

taneous, and this would greatly complicate tlie comparison of g and 
So, were it not for the relative slowness of the molecular motions, 
to which we have already had recourse a moment ago. As it is, we 
may, I think, skip over the difficulty. If A is the distance between 
the molecule P and another Q near it, the interval between the 
moments at which the effective time of P and that of Q have the 

chosen value ^, is of the order of magnitude —^- , as appears from 

(288). The changes which the relative coordinates of Q with respect 
to P undergo during an interval of this length, are of the order 

^^-L\f — , or of the order '%"!- compared with A. The corresponding 

changes in the components of the force' between P and Q are of the 
same order in comparison with the force itself, and may therefore be 

neglected since ^ is very small. In other terms, in order to find 
the force acting on the molecule P, we may consider as simultaneous 
the positions which the surrounding particles occupy at the instants 
at which their local times have the value t\ In virtue of our as- 
sumption, the relative coordinates in these positions bear to x, y , z , 
i. e. to the corresponding coordinates in g^, the ratios determined by 
(286), so that, within the small compass containing P and the mole- 
cules acting on it, the body may be said to have its dimensions 
changed in the way that has often been mentioned. We infer from 
this that the forces acting on corresponding particles, in J>>> and ^q, 
are subjected to the relation (300). 

On the other hand we have the relation (304) between the 
accelerations. Now, if the ratios occurring in (304) und (300) were the 
same, we might conclude that, if the state of motion existing in gg 
is a possible one, so that for each particle the force acting on it is 
equal to the product of its acceleration and its mass, and if the par- 
ticles have equal masses in g and }»lp, the state of motion in the 
former system which corresponds to that in the latter wiU also be 
possible. 

As it is, however, the ratios in (300) and (304) are not equal. 
The above considerations cannot, therefore, lead \is to a theorem of 
corresponding states existing in f«> and gg, unless we give up the 
equality of the masses in these systems. We need not, I think, be 
afraid to make this step. We have seen that the mass of a free 
electron is a function of the velocity, so that, if the corpuscle has 
already the translation iv of the body to which it belongs, the force 
required for a change of the velocity wiU thereby be altered; we 
have further been led to distinguish between a longitudinal and a 
transverse mass. Now that we have already extended to the mole- 
cular interactions the rule that had been deduced for the electric 



LONGITUDINAL AND TRANSVERSE MASS. 205 

forces, it will perhaps not be too rash to imagine an alteration in 
the masses of the molecules caused by the translation, and even, if 
it should prove necessary, to conceive two different masses, one m 
(the longitudinal mass) with which we must reckon when we con- 
sider the accelerations parallel to OX, and another, ot" (the trans- 
verse mass) which comes into play when we are concerned with an 
acceleration, either in the direction of Z or in that of OZ. 

Dividing the ratios in (300) by the corresponding ones in (304), 
we see that, if m^ is the mass of a molecule in the absence of a 
translation, the formulae 



m 
or 



'=(z^:^)mo, m"=(|:,^)wo, 



m' = Tc^lm^, m"=ldm^ (305) 



contain the assumptions required for the establishment of the theorem, 
that the systems g and '^^ can be the seat of molecular motions of 
such a kind that in. both the effective coordinates of the molecules 
are the same functions of the effective time. 

Now, if the molecules of ^g, in their irregular motion, remain 
confined ^vithin a surface having a constant position, those in ^ will 
be continually enclosed by the corresponding surface, i. e. by the one 
that is determined by the same equation in x, y , z . Hence, the 
translation produces the same changes in the dimensions of the 
bounding surface as in those of a body without molecular motions. 

The result may be extended to bodies whose shape and size are 
partly determined by external forces, such as a pressure exerted by 
an adjacent system of molecules, provided only that these forces be 
altered equally with those between the particles of the body itself. 

175. We are now prepared for a theorem concerning correspond- 
ing states of electromagnetic vibration, similar to that of § 162, but 
of a wider scope. To the assumptions already introduced, I shall add 
two new ones, namely 1. that the elastic forces which govern the vibra- 
tory motions of the electrons are subjected to the relation (300), and 
2. that the longitudinal and transverse masses m and m of the 
electrons differ from the mass m^ which they have when at rest in 
the way indicated by (305). The theorem amounts to this, that in 
two systems '^ and g„, the one moving and the other stationary, 
there can be motions of such a kind, that not only the effective co- 
ordinates which determine the positions of the molecules are in both 
the same functions of the effective time (so that the translation is 
attended with the change of dimensions which we have discussed) 
but that the same rule holds for the effective coordinates of the 



206 V. OPTICAL PHENOMENA IN MOVING BODIES. 

separate electrons. Moreover, the components of the vectors d' and 
h' will be found to be identically determined by x , y , s, t', both in 

S and in So- 

In our demonstration we shall regard the displacements of the 
electrons from their positions of equilibrium, and the velocities of 
vibration as very small quantities, the squares and products of which 
may be neglected. We shall also leave aside the resistance that 
may tend to damp the vibrations. 

Let M and Jf^ be corresponding particles of g and gg, and let 
us calculate for these, and for a definite value of the effective time, 
say the value ^', the vector p' whose components are 

p;=Ze^', p;=^e2/', p;=^e/, (306) 

where the sums are extended to all the electrons of the particle con- 
sidered. If we suppose the positions and the motions of the electrons 
to be such as is stated in the theorem, this vector p' will be found 
to be the same for Jf and for Jfp. For the latter particle, p' 
is obviously the electric moment at the time chosen. As to the 
particle Jf, it is to be noticed that if we calculate the sums for the 
chosen value f of the effective time of each electron, the values of 
x, y, z in the sums will not be, strictly speaiiing, the coordinates which 
the different electrons have simultaneously. On account of the small values 
of the vibratory velocities U, we may however simplify the meaning of 
the sums by considering x, y , z as the effective coordinates of the 
several electrons, such as they are at one and the same moment, namely 
the moment when the effective time, taken for a definite point of 
the molecule, which may be called its centre, has the special value t\ 
Then, since the components of the moment of M. at that instant 
may be represented by 

p,= 2:ex„ p,= 2;ej/„ ^=Eez^, (307) 

we shall have, in virtue of (286), 

Px'='«^P:.; P!,'=^P!/; P/=^P.- 

It may be shown that the values of the potentials tp and a' of 
which we have spoken in § 169, are given by the equations, similar 
to (35) and (36), 



^'=I^-jH^'^'^^S', 



where r' is the effective distance between the point P considered and 
a point of the effective element dS'. The square brackets mean that, 
if we wish to determine (p' and a' for the value t' of the effective 



FIELD OP A POLARIZED PARTICLE IN A MOVING SYSTEM, 207 



time, we must understand by q' and u' the values existing in dS' at 
the effective time f ■ 



r 
c 



With the aid of these formulae the electromagnetic field produced 
by a molecule may be shown to be determined in rather a simple 
manner by the vector p', which we may call the effective moment. 
The final formulae, whose form is identical with that of our previous 
equations (271) and (272), are^) 



d' = 



4310* 






Ip:J + _L grad' 



dx r' dy' r' 



d z r 



h' = -^rot' 
iitc 



im\, 



(308) 



where r is the effective distance between the centre of the molecule 
and the point (x', y, z) considered. The square brackets mean that, 
if we want to know the values of d' and h' for the instant at which 
the local time belonging to this point is t', we must take the values 
of p^', p ', p^' for the instant at which the local time of the centre 



of the molecule is t' — 



The dot indicates a differentiation with 



respect to t', and the equations apply as well to the system ^g as 

to 8- 

176. We have next to fix our attention on some molecule M 
of the body j^, and on the one movable electron which we shall 
suppose it to contain. The field produced in M by aU the other 
molecules of the body may be represented by 2Jd' and S\\' (ct § 160), 
but to this we must add the field due to causes outside the body, 
for which the equations are 

0, 
= 0, 



div' do' ■■ 



div' hp 
rot' hn 



= ytlo, 



rot' dg' ■■ 



(309) 



as is seen by putting q = in the formulae (292). 

After having found the total values of d' and h', we can use 
the equations (293), which, however, may be replaced by 



f, = ^^d. 



n= 



I'^V' 



Indeed, so far as d' and h' are due to the vibrations in the other 
molecules of the body, these vectors are proportional to the ampli- 



1) Note 76. 



208 V. OPTICAL PHENOMENA IN MOVING BODIES. 

tudes, so that all the terms in which their components are multiplied 
by uj, Uy' or U/ may be neglected. The corresponding terms with 
components of ('^' and Hq' may likewise be omitted, if the intensity 
of the external field is sufficiently small, if, for example, this field is 
produced by vibrations of very small amplitude in a source of light. 
Returning to the comparison of our two systems, we can finish 
it in few words. On account of what we know of the accelera- 
tions, and of what has been assumed of the masses, it is clear that 
the state of things we have imagined can exist both in g and in g^, 
if all the forces acting on the electrons satisfy the condition (300). 
We have assumed this for the elastic forces, and we can deduce it 
for the electric forces from the equations (310), (308) and (309). 
The effective moments being the same functions of t' in corresponding 
particles of g and go? ^^^ vectors UA' will be so likewise at corre- 
sponding points, and we may suppose the same to be true of the 
vector dp', since one and the same set of equations, namely (309), 
determines it (together with hp'), both for g and for gg. As the com- 
ponents of the force acting on unit charge are given by 6J, d,^', d/ 
for g(,, and by the formulae (310) for g, the condition (300) is 
really fulfilled. 

177. The generalized theorem of corresponding states may now 
serve for the same conclusions which we have drawn from it in its 
more restricted form (§ 165). Attention must, however, be called to 
the difference in frequency between the corresponding vibrations in 
g and 8q. If, for definite values of the eflective coordinates, i. e. 
at a definite point of the system, a quantity varies as cos nf, n will 
be the frequency in the stationary system, because here t' is the true 
time, but for the moving system we shall have 

cosnt' = coanl', t — M—^x), 

so that here the frequency at a definite point of the system is 

I 

h 

It is remarkable that, when the source of light forms part of 
the system, so that it shares the translation w, this frequency wiU 
be produced by the actions going on in the radiating particles, if 
these actions are such that the frequency would be n if the source 
did not move. At least, this is true if we make the natural 
assumption that in the source the masses of the electrons and the 
elastic forces to which they are subjected, are altered in the same 
manner as in a body through which the light is propagated. Then we 
may assert that in the source of light too, the effective coordinates 
of the electrons can be the same functions of the effective time, 



APPLICATION OF THE THBOEEM OF CORRESPONDING STATES. 209 

whether the source move or not. If the vibrations are represented 
in both cases by formulae containing the factor nt', the frequency 

will be n when the source is at rest, and -,-m when it moves. This 

shows that in aU experiments made with a terrestrial source of light, 
the phenomena will correspond quite accurately to those which one 
would observe, using the same source on a stationary planet; the 
course of the relative rays, the position of interference fringes, and, 
in general, the distribution of light and darkness wiU. be unaltered. 
The case of experiments made with a celestial source of light is 
somewhat different. In these, the relative frequency w at a point of 
our apparatus is equal to the frequency of the source, modified accord- 
ing to Doppler's principle (a modification that wiU not exist when we 
employ sunlight, as our distance from the sun may be considered as 
constant), and the phenomena will correspond to those taking place 

with the frequency -^w in a stationary system. Thus, in a dispersive 

medium the courses of the relative rays observed with the 2)-lirieB 
in sunlight and with a sodium flame, would not coincide exactly. If, 
supposing the sun to be at rest relatively to the ether, we caU n 

the relative frequency in the first case, it wiH be r^n m the second 

case. It is scarcely necessary to add that this is of a purely theoretical 
interest, as no phenomenon that can be accurately observed can be per- 

ceptibly altered by this change in the frequency of the order —y- 

It should further be noticed that, in an experiment planned for 
the detection of an influence of the earth's translation, in which we 
turn round our apparatus, or repeat our observations after a certain 
number of hours, during which it has rotated on account of the 
earth's diurnal motion, we are constantly working with the same 
relative frequency (whatever be the source of light employed). This 

constant frequency v will correspond to a determinate frequency y v 

in a system without translation, and the rotation can no more pro- 
duce an effect than it would do if we rotated our instruments on a 
body without translation, on which we were working with rays of 

the frequency y v. 

But perhaps I am dwelling for too long a time upon these 
subtle questions. What must now be pointed out particularly, is, 
that our theorem explains why Rayleigh and Brace have failed to 
detect a double refraction. In the experiments of the latter of these 
physicists the beam of light that was received by the observer's eye 
consisted of two parts, travelling side by side, and having the same 

Lorentz, Theory of electrons. 14 



210 V. OPTICAL PHENOMENA IN MOVING BODIES. 

state of polarization and also, though they had been passed through 
different media, the same intensity. It is clear that, whenever this 
equality exists for two such beams in a system without translation, 
it must, by our theorem, also be found in the corresponding state 
in a moving system. 

178. When, in our comparison of two electrostatic systems g 
and Sq (§ 171), it was stated that, in both of them, the effective 
density of the charge had to be the same function of the effective 
coordinates, this implied that the electrons in the two systems are 
not of the same shape. In the discussion given in § 175, however, 
we have not assumed this, confining ourselves to the two assumptions 
stated in the beginning of that paragraph. Indeed, in dealing with 
the motion of the electrons we are concerned only with their charges, 
their masses and the elastic forces acting on them; aU other par- 
ticulars are irrelevant to our final results. Consequently, we may 
very well conceive the electrons not to change their form and size 
when a body is put in motion (though the dimensions of the body 
itself be altered in the above mentioned manner), provided only that 
the necessary relations between the elastic forces and the masses of 
the electrons, before and after the translation is imparted to the 
system, be maintained. 

Now, in a theory that attempts to explain phenomena by means 
of these minute particles, the simplest course is certainly to consider 
the electrons themselves as wholly immutable, as perfectly rigid 
spheres, for instance, with a constant uniformly distributed surface- 
charge. This is the idea that has been worked out by Abraham, 
and on which many of the formulae I have given in Chap. I are 
based. But, unfortunately, it is at variance with our theorem of 
corresponding states. This requires, as is seen from (305), that the 
longitudinal and the transverse mass of an electron be to each other 
in the ratio 

^ _ 7,2 _ «' 



or, up to quantities of the second order, 

m' .. w^ 

m" e^ ' 

whereas, according to the formulae (6s) and (69), and with the same 
degree of approximation, it would be 

ni 1 _i_ ^ ^^ 

m" 5 c^ 

179. It is for this reason that I have examined what becomes 
of the theory, if the electrons themselves are considered as liable to 



ELECTROMAGNETIC MASS OF A DEPORMABLE ELECTRON. 211 

the same changes of dimensions as the bodies in which they are con- 
tained. This assumption brings out the proper ratios between the 
masses m^, m, m", provided that we assign the value 1 to the coeffi- 
cient I, which we have hitherto left undeterminate. 

The electromagnetic mass of the deformable electron is easily 
deduced from the theory of electromagnetic momentum, since we can 
always apply the general formulae of § 24, whatever be the changes 
in the form and size of an electron taking pLice during its motion. 
By calculating the electromagnetic momentum G and its rate of 
change G, we shall find the force acting on the electron; next, 
when we divide by the acceleration, the electromagnetic mass, either 
the longitudinal or the transverse one, will become known. 

In our calculations we shall ascribe to the electromagnetic mo- 
mentum the value which it would have, if the electron were con- 
tinually moving with the velocity that exists at the moment con- 
sidered, a procedure the legitimacy of which wiU be discussed in a 
subsequent paragraph. 

The determination of the momentum is even more simple than it 
was in the case of a rigid sphere. We have seen that the field of a 
moving electrostatic system is known, when the field of another system 
that is supposed to be at rest, and whose dimensions difi'er in a de- 
finite manner from those of the moving one, is given. Now, if the 
system consists of a single electron, of spherical shape and with 
uniformly distributed surface charge, so long as it stands stiU, but 
ellipsoidal when in motion, as determined by (286), the stationary 
system to the consideration of which the problem is reduced, is found 
to be precisely the original sphere, so that the field is determined 
very easily. 

Calling e the charge, and H the radius of the sphere, T find for 
the electromagnetic momentum corresponding to the velocity w*) 

iG|=-^^-^K (311) 

from which, using the formulae (64) and (65), we deduce 

diklw) „ e^ 



m = 



m" = „ -,^]cL 



6nc^B dw > Gnc^B 

or 

The latter formula agrees with the second of the equations (305), so 
that the only remaining condition is, that the value of m' shall be 
equal to that given by the former of those equations. Hence 

dQclw) ,.,, 
. dw ' 



1) Note 77. 

14* 



212 .V. OPTICAL PHENOMENA IN MOVING BODIES. 

from which, on account of 

dw ' 

we infer 

-, — = 0, Z = const. 

The value of this constant must be unity, because, as we know, 
? = 1 for w; = 0. 

We are thus led so far to specialize the hypothesis that was 
imagined for the explanation of Michelson's experimental result, 
as to admit, for moving bodies, only a contraction, determined by 
the coefficient Jc, in the direction of the line of motion. The elec- 
trons themselves become flattened ellipsoids of revolution, their limiting 
form, which they would reach if the translation had attained the speed 
of light, being that of a circular disk of radius E, perpendicular to 
the line of motion. 

All this looked very tempting, as it would enable us to predict 
that no experiment made with a terrestrial source of light will ever 
show us an influence of the earth's motion, even though it were 
delicate enough to detect effects, not only of the second, but of any 
order of magnitude. But, so far as we can judge at present, the 
facts are against our hypothesis. 

According to it, the longitudinal and the transverse mass of an 
electron would be 

m = k^mg, m" = km^,, 

or, if we put = /5, 

m = (1 - ^^r^l^m^, m" = (1 - /S^)- 1/2^^. (313) 

When /3 becomes greater, these values increase more rapidly than those 
which we have formerly found for the spherical electron. Therefore, 

the determination of —r, for the high velocities existing in the /3-rays 

affords the means of deciding between the different theories. Kauf- 
mann, who, as early as 1901^), had deduced from his researches on 

this subject that the value of ~ increases most markedly, so that 

the mass of an electron may be considered as wholly electromagnetic, 
has repeated his experiments with the utmost care and for the ex- 
press purpose of testing my assumption.^) His new numbers agree 

1) W. Kaufmann, Die magnetisclie und elektrische Ablenkbarkeit der 
Beoquerelstrahlen und die scheinbare Masse der Elektronen, GOtt. Nachr., Math.- 
phys. Kl. 1901, p. 143; tjber die elektromagnetische Masse des Elektrons, ibid. 
1902, p. 291; 1903, p. 90. 

2) Kaufmann, Uber die Konstitution des Elektrons, Ann. Phys. 19 (1906), 
p. 487. 



ENERGY OF AN ELECTRON. 213 

within the limits of experimental errors with the formulae given by 
Abraham, but not so with the second of the equations (313), so 
that they are decidedly unfavourable to the idea of a contraction, 
such as I attempted to work out.^) Yet, though it seems very likely 
that we shall have to relinquish this idea altogether, it is, I think, 
worth while looking into it somewhat more closely. After that, it 
win be weU also to examine a modification of the hypothesis that 
has been proposed by Bucherer and Langevin. 

180. In the preceding determination of the mass of the deformed 
electron we have availed ourselves of the electromagnetic momentum, 
but we have not considered the energy. This was done by Abraham^), 
who found that, besides the ordinary electromagnetic energy, the 
electron must have an energy of another kind, whose amount is 
lessened when the particle is made to move. The truth of this be- 
comes apparent when we consider a rectilinear motion of the electron 
with variable velocity. The mass is given by 

. _ _e^ 73 _ e' (^ w^\-3/2 

and the work of the moving force during the element of time dt by 

whereas the electromagnetic energy is found to be^) 

6nB ' ' 

'Now, the increment of the first term during the time dt is exactly 
equal to the expression (314). 

Hence, there must be another energy E of such an amount that, 
when added to the second term of (315), it gives a constant sum, and 
which is therefore determined by 

where C is a constant. 

181. The nature of this new energy and the mechanism of the 
contraction are made much clearer by the remark, first made by 
Poincare*), that the electron will be in equilibrium, both in its ori- 

1) See, however, Note S7. 

2) M. Abraham, Die Grundhypothesen der Elektronentheorie, Phys. Zeit- 
sehrift 5 (1904), p. 576. 

3) Note 78. 

4) H. Poincare, Sur la dynamique de I'flectron, Rendiconti del Circolo 
matematico di Palermo 21 (1906), p. 129. 






214 V- OPTICAL PHENOMENA IN MOVING BODIES. 

ginal and in its flattened form, if it has the properties of a very 
thin, perfectly flexible and extensible shell, whose parts are drawn 
inwards by a normal stress, having the intensity 

-9. 

s = 



per unit of area, and preserving this magnitude however far the con- 
traction may proceed. 

The value of S has been so chosen that, so long as the electron 
is at rest and has therefore the shape of a sphere with radius B, 
the internal force exactly counterbalances the electromagnetic stress 
on the outside which is due to the surrounding field.^) Now — and 
herein lies the gist of Poincare's remark — the electron, when 
deformed as has been stated, will still be in equilibrium under the 
joint action of the stress S and the electromagnetic forces. 

In order to show this, we shall fix our attention on the com- 
ponents of the internal stress acting on a surface element of the 
shell; these are found if we multiply S by the projections of the 
element on the planes of ys, zx, xy. Now, when the deformation 

takes place, these projections are multiplied by the factors 1, -^, -^ , 

from which it appears that the components of the stress are altered 
in the same ratios as those of the electromagnetic force (cf. (300)), so 
that the equilibrium will still persist. When it is stable, the electron 
will necessarily have the configuration corresponding to it; the electro- 
magnetic forces exerted on its surface by the ether, modified by 
the translation according to our formulae, conjointly with the in- 
variable internal stress, will make the electron take the flattened 
ellipsoidal form. 

Corresponding to the internal stress S there must be a certain 
potential energy U, and the above result implies that this energy is 
equal to the expression (316). Indeed, if v is the volume of the 
ellipsoid, we obviously may write 

U = Sv -\- const = ^ — Y^.v + const, 
and we have 

\l/2 






182. Abraham^) has raised the objection that I had not shown 
that the electron, when deformed to an ellipsoid by its translation, 
would be in stable equilibrium. This is certainly true, but I think 
the hypothesis need not be discarded for this reason. The argument 

1) Note 79. 

2) Abraham, 1. c, p. 578. 



STABILITY OP THE STATE OE AN ELECTRON. 215 

proves only that the eleotromagrietic actions and the stress of which 
we have spoken cannot be the only forces which determine the 
configuration of the electron. 

If they were, each problem concerning the relative motion of 
the parts of the moving ellipsoidal electron would have its counter- 
part in a problem relating to the spherical electron without trans- 
lation, because the forces of both kinds would satisfy the relation (300). 
Now, it is easily seen that, under the joint action of the stresses in 
the surrounding field and the constant internal stress S, a spherical 
shell would be in stable equilibrium as regards changes of volume, 
but that its equilibrium would be unstable with respect to changes 
of shape. 1) The same would therefore be true of the moving and 
flattened shell. In the case of the latter there would even be in- 
stability of orientation, because after a small rotation the electron 
does no longer correspond [after the manner indicated by the for- 
mulae (286)] to the original sphere, but to a slightly deformed one. 

Notwithstanding aU this, it would, in my opinion, be quite 
legitimate to maintain the hypothesis of the contracting electrons, if 
by its means we could reaUy make some progress in the understanding 
of phenomena. In speculating on the structure of these minute par- 
ticles we must not forget that there may be many possibilities not 
dreamt of at present; it may very well be that other internal forces 
serve to ensure the stability of the system, and perhaps, after all, 
we are wholly on the wrong track when we apply to the parts of 
an electron our ordinary notion of force. 

Leaving aside the special mechanism that has been imagined by 
Poincare, we are offered the following alternative. Either a spheri- 
cal electron must be regarded as a material system between whose 
parts there are certain forces ensuring the constancy of its size and 
form, or we must simply assume this constancy as a matter of fact 
which we have not to analyze any further. In the first case, the 
form, size and orientation of the moving ellipsoid will also be 
maintained by the action of the system of forces, provided aU of 
them have the property expressed in our relation (300). In the other 
case we may rest content with simply admitting for the moving 
electron, without any further discussion, the ellipsoidal form with 
the smaller axis in the line of translation. 

183. I must also say a few words about another question that 
is connected with the preceding one. In our calculation of the 
masses m' and m" in § 179 we have assumed that at any moment 
the electromagnetic momentum has the value corresponding, in a 



1) Note 80. 



216 V. OPTICAL PHENOMENA IN MOVING BODIES. 

stationary state of motion, to the actual velocity. Particularly, in the 
application of the formula (311), it has been presupposed that in a 
curvilinear motion the electron constantly has its short axis along 
the tangent to the path, and that, while the velocity changes, the 
ratio between the axes of the ellipsoid is changing at the same time. 

Strictly speaking, it is not absolutely necessary for our results 
that the orientation and shape of the electron should follow in- 
stantaneously the alterations in direction and velocity of its trans- 
lation; they may be supposed to lag somewhat behind. But it is 
clear that, at all events, if we want to apply the values of m and 
m" to optical phenomena, as we have done, the time of lagging must 
be small in comparison with the period of the vibrations of light. 

Now, if we choose the latter of the alternatives that presented 
themselves in the last paragraph, we may as well simply assert that 
there is no lagging at all. But we must not proceed in this summary 
manner if we prefer the first alternative. If the form and the orien- 
tation of the electron are determined by forces, we cannot be certain 
that there exists at every instant a state of equilibrium. Even while 
the translation is constant, there may be small oscillations of the 
corpuscle, both in shape and in orientation, and under variable 
circumstances, i. e. when the velocity of translation is changing either 
in direction or in magnitude, the lagging behind of which we have 
just spoken cannot be entirely avoided. The case is similar to that 
of a pendulum bob acted on by a variable force, whose changes, as 
is weU known, it does not instantaneously foUow. The pendulum 
may, however, approximately be said to do so when the variations 
of the force are very slow in comparison with its own free vibra- 
tions. Similarly, the electron may be regarded as being, at every 
instant, in the state of equilibrium corresponding to its velocity, pro- 
vided that the time in which the velocity changes perceptibly be very 
much longer than the period of the oscillations that can be performed 
under the influence of the regulating forces. If these vibrations are 
much more rapid than those of light, the values (313) of the masses 
m' and m" may be confidently applied to the electrons in a body 
traversed by a beam of light, and with even more right to free elec- 
trons that are deflected from their line of motion by a magnetic or 
an electric field. 

Of course, since we know next to nothing of the structure of 
an electron, it is impossible to form an opinion about the period of 
its free oscillations, but perhaps we shall not be far from the mark 
if we suppose it to correspond to a wave-length of the same order 
of magnitude as the diameter. 

It appears from these considerations that the idea of a deforma- 
bility of the electrons would give rise to several new problems. One 



EIGID SPHERICAL ELECTRONS. 217 

of these would be that of the rotation of these particles. An electron 
is set spinning whenever a magnetic force to which it is exposed 
undergoes a change, and it would be necessary to obtain an insight 
into the peculiarities of the motion imparted in cases of this kind 
to our flattened ellipsoids. 

184. As has already been observed (§ 178\ the often mentioned 
changes in the internal forces, and consequently in the dimensions 
of a body can be imagined without extending the assumption to the 
electrons themselves and the question therefore naturally arises, 
whether after all we may not get a satisfactory theory by simply 
adhering to the idea of rigid sphei'ical electrons. This course would 

be open to us, if the discrepancy between the values of „ given at 

the end of § 178 could be shown to have no perceptible influence 
on observed phenomena. In examining this point we are led back 
to the question of double refraction of which we have already 
spoken. 

A glance at the formulae that have served us in Chap. IV for 
treating the propagation of light in a system of molecules, shows 

that the term m'n^ = ^— | in equation (201) is the only one which 

contains the mass of an electron. Moreover if, confining ourselves 
to perfectly transparent bodies (not subjected to an external magnetic 
force), we leave aside the resistance to the vibrations, that term 
is also the only one in which there is any question of the frequency. 
It follows that all depends on the product mn^, and that a change 
of in, say in the ratio of 1 to a, will have the same effect as a 
change of n in the ratio of 1 to «'/*. 

Let us now suppose for a moment that the values of the two 
masses of an electron, though not exactly equal to the expressions (313), 
are at least proportional to these, say 

m' = aWm^, m" = aJcm^, (317) 

where the coefficient « is a certain function of the velocity of trans- 
lation tr, equal to unity for w = 0, and differing from 1 by a quantity 
of the second order when w is small. Then, the phenomena in a 
moving system ^ and those in a stationary one ^^ will correspond 
to each other as formerly explained, provided that the mass of the 
electrons in the system go be not m,^ but um,^. If the body considered 
were originally isotropic, a change of the mass of its electrons from m^ 
to am^ certainly would not make it doubly refracting. Hence, the 
moving body whose electrons have the masses (317), can be so 
neither. It must be singly refracting, and we may be sure that 
practically it wiU present the same phenomena, in experiments, that 



218 V. OPTICAL PHENOMENA IN MOVING BODIES. 

is, in which the source of light moves with it, as it would do when 
kept at rest. It is true that there wiU be a difference equivalent to 
that which could be caused by a change of the mass of an electron 
from JWg to aniQ, or by one of the frequency to a corresponding 
amount (of the second order), but certainly this can have no per- 
ceptible influence. 

We shall next consider the case that the longitudinal and the 
transverse mass of an electron bear to each other a ratio different 
from ]c^. Let us write for their values 

m' = h'm^, m" = h"m^, 

where li and h" are factors having similar properties as the above 
coefficient a. Then the phenomena in the body ^ correspond to 
those in a body gj in which the electrons would have a mass 

V 

with respect to accelerations parallel to OX, and a mass 

h" 

with respect to accelerations at right angles to that line. A body 
of this kind would undoubtedly show a double refraction, and so 
would the moving body g corresponding to it. If, for example, a 
ray of light were propagated along a line perpendicular to OX, say 
in the direction of Y, the velocity of propagation would be different 
according as the vibrations were parallel to OX or to OZ. The 
frequency of the light used being denoted by n, the velocity of pro- 
pagation of one vibration would be (by the theorem with which we 
have begun this paragraph) as if the frequency were 

and that of the other as if it were 

the masses being taken equal to m^ in both cases. 

185. For a spherical electron we have, according to the for- 
mulae (70) and (71), if we neglect terms of an order higher than 
the second, 

A'=i + |^^, r= 1 + 1^2 

and, as we may put 

the above values become 

(l-|,/^> and (l-4/3^)n, 



SEARCH FOR DOUBLE REFRACTION CAUSED BY TRANSLATION. 219 
showing a difference of 

since the velocity of the earth is one ten-thousandth part of the 
speed of light. In the case of water, and for yellow light, this 
change of frequency would produce a change in the index of refrac- 
tion of about 2 10~ii, and this, therefore, would be the difference 
between the two principal refractive indices which we might expect 
ia the double refraction experiment. 

It is scarcely necessary to say that Rayleigh's^) and Brace's^) 
observations were conducted in such a manner that a double refrac- 
tion in which one of the principal directions of vibration would be 
parallel to the earth's motion could manifest itself. As I mentioned 
already, the results were invariably negative, though Brace's means 
of observation were so sensitive that a difference between the prin- 
cipal refractive indices of 10~^^ could not have escaped him. This is 
about the twentieth part of the value which we have just computed. 

It is true that we have based our calculations on certain 
assumptions that could be changed for others, and Brace himself 
has made the calculation in a different manner. Yet, I think, we 
may confidently conclude that it will be extremely difficult to reconcile 
the result of his observations with the idea of rigid spherical electrons. 

It must be added that, if we adhered to this idea, our con- 
siderations concerning the molecular motions in a moving system 
would also require some modification. 

186. We have seen in § 184 that there would be no contradic- 
tion with Brace's results, if the ratio between the longitudinal and 
the transverse mass had the value Jc^. This raises the question as 
to whether this latter ratio can be obtained without the assumption 
1=1, which has been the origin of aU our difficulties. Unfortunately, 
this way out is barred to us, the equation 

d{klw) 



dw 



11 = 



being satisfied only by a constant value of I. 

For this reason the optical experiments do not allow us to 
suppose, as has been done by Bucherer') and Langevin*), that a 

1) Rayleigh, Does motion through the aether cause double refraction? 
Phil. Mag. (6) 4 (1902), p. 678. 

2) D. B. Brace, On double refraction in matter moving through the aether, 
Phil. Mag. (6) 7 (1904), p. 317. 

3) A. H. Bucherer, Mathematische Einfiihrung in die Elektronentheorie, 
Leipzig, 1904, p. 57 a. 58. 

4) P. Langevin, La physique des flectrons. Revue gen&ale des sciences 
pures et appliqnees 16 (1905), p. 257. 



220 V. OPTICAL PHENOMENA IN MOVING BODIES. 

moving electron is deformed to an ellipsoid of the form and orien- 
tation which I have assigned to it, but having the original volume, 
instead of the original equatorial radius. This assumption obviously 
amounts to putting I = li~ ^1^ , so that the dimensions of the electron 
would be altered in the ratios h"^!^ , W^, W^. When we use this 
value of I, the two electromagnetic masses become 

m'=(l-/3r^/Hl-i/30»^o, (318) 

m _ 1— 1(3^ 



giving for the ratio 



instead of 



m" \ — ^^> 



If we apply to this hypothesis the same mode of calculation as 
to that of rigid spheres, we are led to a double refraction that is 
even a little stronger, so that the contradiction with Brace's experi- 
ments would remain the same. 

This is certainly to be regretted as the new assumption has 
unmistakeable advantages over my original one. It is in sufficient 
agreement with Kaufmann's results, and the idea of a constant vo- 
lume is indeed very simple. Following it we should not be obliged, as 
we were in § 180, to admit the existence of another energy than the 
ordinary electromagnetic one. This is confirmed by the magnitude 
of the electromagnetic energy^) 

and the expression 

derived from (318), for the work of the force, in case the electron 
has a rectilinear motion of variable velocity. The latter quantity is 
equal to the increment of the former in the time dt. 

187. It is interesting, now to turn once more to the hypo- 
thesis 1=1, combined with the formulae (305) for the masses (assu- 
ming as a matter of fact the influence of the translation on the 
masses expressed by these equations), and to consider the equations 

1) Note 81. 



GENERAL ELECTROMAGNETIC EQUATIONS FOR A MOVING SYSTEM. 221 



for the propagation of light in moving transparent bodies to which 
it leads. We have seen that the vectors d', h', p' can be the same 
functions of x' , y , z', t' both in a moving system g and in a statio- 
nary one §0. The same must be true of some other vectors that 
can be derived from them, viz. 1. the vector E' which we define as the 
mean value d' of d', taken, in g^ for a spherical space, infinitely 
small in a physical sense, with its centre at the point considered, 
and in g for the space corresponding to that sphere, 2. the mean 

value h' defined in the same way, and to be denoted by H', 3. the 
vector 

V = N'f,', (319) 

where, in the formulae for both systems, we understand by N the 
number of molecules which ^^ contains per unit of volume, and 
4. a vector D' defined by the equation 



D = E'+P' 



(320) 



Since all these vectors can be, in §„ and in g, the same functions 
of x, y', z', t', the equations by which they are determined must be 
such that they can be written in the same form. 

Now, for the system gg, x, y, z , t' are the true coordinates and 
the true time, whereas the above vectors are what we formerly caUed 
E, H, P and D. As we know that they satisfy the equations 



(321) 



div D = 0, 




div H = 0, 




rot H = - 

c 


dt ' 


rot E = - 


1 aH 

c dt 



we may be sure that, for the moving system, 

div' D' = 0, 
div' H' = 0, 

,, „, 1 ao' 

rot H = — r.,,- , 
c dt ' 



rot' E' = - 



1 cW 



(322) 



where the symbols div' and rot' have the meaning that has been ex- 
plained in § 169. 

To (321) must be added the relation between E and D, and to 
(322) a corresponding relation between E' and D', so that, if we write 

D = F(L), (323) 



222 



V. OPTICAL PHENOMENA IN MOVING BODIES. 



we shall also have 



FiEy 



(324) 



Here, the symbol F must be understood in a very general sense; 
it is meant to include aU forms which the equations may take 
according to the special properties of the body considered. If the 
first formula contains, as may very weU be'), differential coefficients 
with respect to t, we shall find in the second the corresponding 
differential coefficients with respect to t'. 

Putting D = E, and similarly D' = E', we obtain the equations 
for the free ether. These, however, may be left in the form 



div d = 0, 
div h = 0, 

1 ad 



rot h = 



'dt 



, . 1 ah 

rot d = KT 

e dt 



(325) 



div' d' 


= 0, 




div' h' 


= 0, 




rot' h' 


1 
c 


ad' 


rot' d' 


= — 


1 ah' 

c dt' 



for the system gj, there being no necessity for considering mean 
values when there are no molecules, and we may write for them 



(326) 



when we are concerned with >^. 

As the ether does not share the translation w, the two last sets 
of equations serve for exactly the same phenomena. The one is 
derived from the other by purely mathematical transformations, the 
only difference between the two being, that the electromagnetic field 
is referred to axes fixed in the ether and to the „true" time in (325), 
but to moving axes and „local" time in (326), and that it is de- 
scribed in the two cases by means of different vectors. On the con- 
trary, the phenomena to which the equations (321, 323) and (322, 
324) apply, though corresponding to each other, cannot be said to be 
identical. 

188. Having got thus far, we may proceed as is often done in 
theoretical physics. We may remove the scaffolding by means of 
which the system of equations has been built up, and, without 
troubling ourselves any more about the theory of electrons and the 



1) Note 82. 



EINSTEIN'S THEORY. 223 

difficulties amidst which it has landed us, we may postulate the 
above equations as a concise and, so far as we know, accurate de- 
scription of the phenomena. From this point of view, E, H, D 
in one system, and E', H', D' in the other, are simply „certain" 
vectors, about whose meaning we say just so much as is necessary 
for fixing unequivocally for every case their directions and magni- 
tudes. 

As to the grounds on which the equations recommend them- 
selves, these are: 1. that the formulae (321), combined with suitable 
assumptions concerning the relation between E and D, can serve for 
the explanation of optical phenomena in transparent bodies, whether 
singly or doubly refracting, 2. that the identity in form of (321, 323) 
and (322, 324) accounts for the failure of all attempts to discover 
an influence of the earth's motion by experiments with terrestrial 
sources, and 3. that the equations (322, 324) give the right value 
for Fresnel's coefficient. 

189. The denominations „effective coordinates", „effective time" 
etc. of which we have availed ourselves for the sake of facilitating 
our mode of expression, have prepared us for a very interesting 
interpretation of the above results, for which we are indebted to 
Einstein.^) Let us imagine an observer, whom we shall call A^ and 
to whom we shall assign a fixed position in the ether, to be engaged 
in the study of the phenomena going on in the stationary system g^. 
We shall suppose him to be provided with a measuring rod and a 
clock, even, for his convenience, let us say, with a certain number 
of clocks placed at various points of ^q, and adjusted to each other 
with perfect accuracy. By these means he will be able to determine 
the coordinates x, y, z for any point, and the time t for any instant, 
and by studying the electromagnetic field as it manifests itself at 
different places and times, he wiU be led to the equations (321, 323). 

Let J. be a second observer, whose task it is to examine the 
phenomena in the system ^, and who himself also moves through 
the ether with the velocity w, without being aware either of this 
motion or of that of the system ^. 

Let this observer use the same measuring rod (or an exact copy 
of it) that has served A^, the rod having acquired in one way or 
another the velocity w before coming into his hands. Then, by our 
assumption concerning the dimensions of moving bodies, the divisions 
of the scale will in general have a length that differs from the ori- 



1) See Ann. d. Phys. 17 (1905), p. 891; 18 (1905), p. 639; 20 (1906), p. 627; 
21 (1906), p. 583; 23 (1907), p. 197, 371, and the comprehensive exposition of 
Einstein's theory.- tJber das Relativitatsprinzip und die aus demselben ge- 
zogenen Folgerungen, Jahrb. d. Kadioaktivitiit u. Elel?tronik 4 (1908), p. 411. 



224 V. OPTICAL PHENOMENA IN MOVING BODIES. 

ginal one, and will even change whenever the rod is turned round, 
the law of these changes being, that, in corresponding positions in 
^0 and ,S, the rod has equal projections on the plane YOZ, but 
projections on OX whose ratio is as ]c to 1. It is clear that, since 
the observer is unconscious of these changes, he will be unable 
to measure the true relative coordinates x^ of the points of the 
system. His readings will give him only the values of the effec- 
tive coordinates x' and, of course, those of y', z which, for l=\, 
are equal to y^, z^. Hence, relying on his rod, he will not find the 
true shape of bodies. He wiU take for a sphere what really is an 
ellipsoid, and his cubic centimetre will be, not a true cubic centi- 
metre, but a parallelepiped /c times smaller. This, however, contains 
a quantity of matter, which, in the absence of the translation, would 
occupy a cubic centimetre, so that, if A counts the molecules in Ms 
cubic centimetre, he will find the same number iV as ^q. Moreover, his 
unit of mass wiU be the same as that of the stationary observer, if 
each of them chooses as unit the mass of the water occupying a 
volume equal to }ds cubic centimetre. 

With the clocks of A the case is the same as with his measu- 
ring rod. If we suppose the forces in the clock-work to be liable 
to the changes determined by (300), the motion of two equal clocks, 
one in ,S„ and the other in [->, will be such that the effective coordi- 
nates of the moving parts are, in both systems, the same functions 
of the effective time. Consequently, if the hand of the clock in Qq 
returns to its initial position after an interval of time ©, the hand 
of the clock in ,S 'w^iU do so after an increment equal to of the 
effective time f. Therefore, a clock in the system [^ will indicate 
the progress of the effective time, and without his knowing anything 
about it, J.'s Clocks wiU go li times slower than those of A^^. 

190. It foUows from what has been said that, if the moving 
observer measures the speed of light, by making a ray of light travel 
from a point P to a point Q, and then back to P, he will find the 
value c. This may be shown for every direction of the line PQ^), 
but it wiU suffice to give the proof for the case that the line is 
either parallel to OX, or at right angles to it. If L is the distance 
between P and Q as measured by A, then in the first case the true 

distance is y, and, as both points move through the ether with the 

velocity w, the time required by the ray of light is 

- (--- + -' -^ = -^- - = ^^ • r327^ 



1) Note 83. 



MEASUREMENTS IN A MOVING SYSTEM. 225 

In the second case the light has to travel along two sides of an 
isoscele triangle (cf. § 167), whose height is L and whose half base is 
to one of the sides as tv to c. The side is therefore 



V 






and the time taken by the beam of light to return to its starting 
point is again given by (327). As ^'s clock goes h times too slow, 

it will mark an interval of time " , so that the observer wiR con- 

elude that the velocity of the rays is equal to c. 

Let us now suppose him to be provided with a certain number 
of clocks placed at different points of his system, and to adjust these 
clocks to each other by the best means at his disposal. In order to 
do so with two clocks placed at the points P and Q, at a measured 
distance L from each other, he may start an optical signal from P 
the moment at which the first clock marks the time t' = 0, and 
so set the second clock that, at the arrival of the signal, it marks 

the time — , making allowance in this way for the time of passage 

of the light which he judges to be -~ • 

Let us suppose that P lies at the origin of coordinates, and Q 
on the positive axis of x; further, that a clock without translation and 
therefore indicating the true time, marks the instant at the moment 
of signalling. Then, on account of the different rates of a moving 
and a stationary clock, we shall have continually for the clock at P 

K 

At the moment of arrival of the signal the true time wiU be 

L 

k(c — IV) ' 

since this is the interval required for the passage of the light be- 
tween the points P and Q, which move with the velocity w and 

whose true distance is -y- ■ 
k 

Now, since at this moment the time indicated by the clock at Q 
is — , its indication, at any other true time t, will be 



*'" c + /fc r lc{c-w)]> 



or, since L = x 

Lorentz, Theory of electrons. 15 



,, _ 1 . W , 



226 V. OPTICAL PHENOMENA IN MOVING BODIES. 

This agrees exactly with (288), so that we see that when the 
clocks are adjusted by means of optical signals, each of them wiU 
indicate the local time t' corresponding to its position. 

The proof may easily be extended to other directions of the 
line joining the two places.') 

191. It is of importance not to forget that, in doing all that 
has been said, the observer would remain entirely unconscious of 
his system moying (with himself) through the ether, and of the 
errors of his rod and his clocks. 

Continuing his researches he may now undertake a study of the 
electromagnetic phenomena in his system, in exactly the same manner 
in which A^ has done so in his. We can predict what his results 
will be, because we know the phenomena by our theorem of corre- 
sponding states. From this we can infer that, if the moving observer 
determines velocities and accelerations in terms of his effective co- 
ordinates and his effective time, if he deduces the intensity of forces 
from the acceleration which they give to unit of mass, and if he 
measures electric charges in the ordinary way by means of the electro- 
static actions which they exert on each other, his unit of electricity 
win be equal to that chosen by A^. His density of charge, on the 
contrary, wiU not be the true density q, but what we have caUed 
the effective density q. Further, if he determines the force acting 
on unit charge at some point of the electromagnetic field, he will 
find the vector d'.-) Similarly he will be led to consider the vector h', 
and, pursuing his study, he will sooner or later come to establishing 
the equations that determine the field, namely the formulae (326) for 
the free ether and (322, 324) for a ponderable body. 

He may attain this latter object by different courses. Perhaps he 
will be satisfied with the idea that D' is a certain vector which he has 
for the first time occasion to introduce when working: with a charged 
condenser. Or, if he develops a theory of electrons, he will get the 
notion of the electric moment of a particle, whose components he 
will naturally define by the expressions '^ex', "^cy, ^e/, so that 
what he calls the moment is in reality the vector p' of our equa- 
tions (306). After having introduced it, the moving observer will 
define P' and D' by the formulae (319) and (320). 

We may sum up these considerations by saying that, if both ^^ 
and A were to keep a record of their observations and the con- 
clusions drawn from them, these records would, on comparison, be 
found to be exactly identical. 

192. Attention must now be drawn to a remarkable reciprocity 
that has been pointed out by Einstein. Thus far it has been the 

1) Note 84. 2) Note 85. 



EINSTEIN'S THEORY. 227 

task of the observer A^ to examine the phenomena in the stationary 
system, whereas A has had to confine himself to the system g. Let 
us now imagine that each observer is able to see the system to which 
the other belongs, and to study the phenomena going on in it. Then, 
Ag will be in the position in which we have all along imagined 
ourselves to be (though, strictly speaking, on account of the earth's 
motion, we are in the position of A)-, in studying the electromagnetic 
field in g, he wiU be led to introduce the new variables x', y , /, 
d', h', etc. and so he wiU establish the equations (326) and (322, 324). 
The reciprocity consists in this that, if the observer A describes in 
exactly the same manner the field in the stationary system, he will 
describe it accurately. 

In order to see this, we shall revert to the equations (287) and 
(288), which in our present hypothesis I = 1 take the form 

x =l{x — ivi), y'=y, z'=z, i ='k\t—^_x\. (328) 

Let P be a point belonging to the system ^q and let us fix 
our attention on the coordinate x which it has with respect to the 
moving axes of |^, for two definite values t' and f + Ai' of the 
local time. Since x is constant for this point P, we have by the 
last of the above equations 

K 

and by the first 

Aa;' = — hwtit = — w£^t'. 

Judging by bis means of observation, the observer A wiH therefore 
ascribe to the system ^f, a velocity w in a direction opposite to that 
of the positive axis of x' . 

Just as Af^, in his theory of the electromagnetic field in '^, has 
changed the coordinates x, y, z, the time t and the electromagnetic 
vectors d, h, E, H, P, D for the variables (328), the vectors d', h', 
whose components are 



K-K, d; = ^(d,-fh,), d; = ^d,+ ^h,). 



(329) 



and the vectors E', H', P', D', so the observer A will introduce, in- 
stead of the quantities x, y , z, t', d', etc. which belong to his system, 
certain new quantities which we shall distinguish by double dashes, 
and which will serve him in his theory of the system S„. 

He win define these new quantities by equations analogous to 
(328) and (329), replacing w by — ir, which however does not affect, 
the constant h. His transformation will therefore be as foUows 

1.5* 



(330) 
(331) 



228 "V. OPTICAL PHENOMENA IN MOVING BODIES. 

x"=li{x' + wt'), y" = y, /' = /, t"==lc{t' + ^,x), 

If he also defines the vectors E", H", D" similarly to A^s definition 
of E', H', D', the observer A will finally find the following equations, 
to be applied to the system go; and corresponding to (326), (322, 
324); for the ether 

div"d"=0, 

div"h"=0, 

rot n = — „-7r , 
c dt 

.,, J" 1 9h" 

rot d = Krrr , 

c ot 
and for a ponderable body 

divD"=0, 
divH"=0, 

1 3D" 
V dt" ' 

l^dW" 

c > t" ' 

D" = ^-(£'0 
The symbols div" and rot" will require no further explanation. 



(332) 



rot" H" 
rot"E" = 



(333) 



(334) 



193. It* remains to show that these formulae contain an accurate 
description of the phenomena in ^{,. The proof of this is very 
simple, because, if we look at them somewhat more closely, the 
equations are found to be the same which ^^ has used for the 
purpose. 

Indeed, if we solve x, y, s, t from the equations (328) and d^, 
d , d^, h^, h , hj from (329), we find values agreeing exactly with 
(330) and (331), so that 

x" = x, y" = y, z"=z, t"=t, 

d"=d, h"=h, 

by which the identity of the sets of equations (332) and (325) is 
demonstrated. As to the equations (333, 334) and (321, 323), the 
only difference between the two sets is, that one contains the vectors 
E" and D", and the other the vectors E and D. If these four quan- 
tities are considered simply as „certain" vectors (represented by 



PRINCIPLE OP RELATIVITY. 229 

symbols the choice of which is immaterial), this similarity in form, 
together with our knowledge that in free ether E"=E, D"= D, 
H"=H (since for this medium E"=D"=d", E = D = d, H"=h"^ 

H = h) must, and can, suffice for our conclusion that the phenomena 
in the system S„ can be described by means of the equations (333, 
334) just as weU as by (321, 323). 

We may go a step farther if we suppose that the moving and 
the stationary observer, or rather theorist, as they have now become, 
establish a theory of molecules and of electrons. A^ has defined 
E', H' as the mean values of d', h', and for the other vectors he has 
used the equations 

D' = E' + P'. 

Similarly, A wiU define E" and H" as the mean values of d" and h", 
so that these vectors become equal to the mean values of d and h, 
i. e. to E and H. He will put for each molecule 

and further 

P" = JVp", 

D ' = E + F. 

Comparing these formulae with (307) (for which we may write 
p^ = '^ex, etc.) and the equations P = iVp, D = E + P, and keep- 
ing in mind that x" = x, y"=y, z"=z, we see that reaUy 
p"=p, P' = P, D'=D. 

194. It will be clear by what has been said that the impressions 
received by the two observers A^ and A would be alike in aU re- 
spects. It would be impossible to decide which of them moves or 
stands still with respect to the ether, and there would be no reason 
for preferring the times and lengths measured by the one to those 
determined by the other, nor for saying that either of them is in 
possession of the „true" times or the „true" lengths. This is a point 
which Einstein has laid particular stress on, in a theory in which 
he starts from what he calls the principle of relativity, i. e. the prin- 
ciple that the equations by means of which physical phenomena may 
be described are not altered in form when we change the axes of 
coordinates for others having a uniform motion of translation rela- 
tively to the original system. 

I cannot speak here of the many highly interesting applications 
which Einstein has made of this principle. His results concerning 
electromagnetic and optical phenomena (leading to the same contra- 



230 V. OPTICAL PHENOMENA IN MOVING BODIES. 

diction with Kaufmann's results that was pointed out in § 179) agree 
in the main with those which we have obtained in the preceding 
pages, the chief difference being that Einstein simply postulates 
what we have deduced, with some difficulty and not altogether satis- 
factorily, from the fundamental equations of the electromagnetic field. 
By doing so, he may certainly take credit for making us see in the 
negative result of experiments like those of Michelson, Rayleigh 
and Brace, not a fortuitous compensation of opposing effects, but 
the manifestation of a general and fundamental principle. 

Yet, I think, something may also be claimed in favour of the 
form in which I have presented the theory. I cannot but regard the 
ether, which can be the seat of an electromagnetic field with its 
energy and its vibrations, as endowed with a certain degree of sub- 
stantiality, however different it may be from aU ordinary matter. In 
this line of thought, it seems natural not to assume at starting that 
it can never make any difference whether a body moves through the 
ether or not, and to measure distances and lengths of time by means 
of rods and clocks having a fixed position relatively to the ether. 

It would be unjust not to add that, besides the fascinating 
boldness of its starting point, Einstein's theory has another marked 
advantage over mine. Whereas I have not been able to obtain for 
the equations referred to moving axes exactly the same form as for 
those which apply to a stationary system, Einstein has accomplished 
this by means of a system of new variables slightly different from 
those which I have introduced.^) I have not availed myself of his 
substitutions, only because the formulae are rather complicated and 
look somewhat artificial, unless one deduces them from the principle 
of relativity itself^) 



1) Note 86. 2) Note 87. 



NOTE 8.^) 

1 (Page 6). Equation (4) is equivalent to the three formulae 

^ _ ^ = A y- 

dy dz c dt ' 

dz dx e dt ' 
dhy dh^ _ 1 3dj 
dx dy e dt 

When the second of these, differentiated with respect to 2, is 
subtracted from the third, differentiated with respect to y, we find 

L /^.r <Shj,dK\_.. _ 1 A M _ ^<^A ri) 

dxKdx'^ dy^ cs) ^ c dtXdy dz)' ^^^ 

or, if (3) and (5) are taken into account, 

Ah =i^^. 
^"^ c' dt^ 

Corresponding formulae for hj^, h. and for the components of d 
are obtained in a similar manner. 

It may be noticed that the quantity 

^ /Shy _ dh^\ _ d_ tdj\^ _ ah,\ 
dy\dx dy) dz\dz dx)' 

which we have calculated in the above transformation, is the first 
component of the rotation of rot h, or, as we may say, of rot rot h, 
and that the expression on the left-hand side of (I) is the first com- 
ponent of the vector 

grad div h — Ah. 

In general, denoting by A any vector, we may write 

rot rot A = grad div A — AA, (5) 

a theorem which enables us to perform in the terms of vector ana- 
lysis the elimination of d from the fundamental equations. Indeed, 
we may deduce from (4) 

rot rot h = - rot d , 



1) The numbers of the formulae in this Appendix will be printed in italics. 



232 NOTES. 



• Pi 

rot d = -oT rot d , 



1 /) 

grad div h — A h = — -^ rot d, 



i. e., if we use (3) and (5) 
Similarly, the equation 






AH 1 S't" 

is obtained it we begin by considering the vector rot rot d. 

2 (Page 16). The definitions given in § 2 lead to the general 
formula 

div rot A = 0. 

Hence the equation (19) requires that 

divc = div(d + 9V) =0, (5) 

i. e. that the total current, which is composed of the displacement 
current d and the convection current qW , be solenoidally distributed. 
In order to show that it is so whenever the condition mentioned in 
the text is fulfilled, we shall fix our attention on an element of the 
charged matter, situated at the time t at the point (x, y, s), and 
therefore, at the time t + dt, at the point {x -\- \^dt, «/ + V, dt, 
z + \^dt). By a well known theorem of the theory of infinitely 
small deformations, the volume of the element, if initially equal to dS, 
will have become 

+ (£+l?+S)'»l''« w 

at the end of the interval dt. 

On the other hand, the time having changed by dt and the 
coordinates by \^dt, V^^dt, V.dt, the density of the charge, which at 
first was Q, has become 

P + (-|7 + V.|^ + ^--+V-|^)^^. 
\ dt "^ Sx ' V cy ' - dz I 

The product of this expression by {4) must be equal to the 
original charge QdS of the element, so that we have 

^\dx ^ dy^ ds) ^ dt ^"^dx ^^ydy +^-^~"' 
or 

]| + div(pv) = 0, (5) 



3, 4 NOTES. 233 

from which, taking into account (17), we are at once led to the 
equation (5). 

3 (Page 17). The method of elimination is exactly like that 
which we used in Note 1. We may infer from (20) and (19) 

rot rot d = rot h , 

rot rot h = - rot d H rot (pv), 

or, using (5), 

1 rl 

grad divd — Ad= -g- (rot h), 

grad div h - Ah = 1 1^ (rot d) + -^ rot (pv), 

and we get the formulae (24) and (25), if we substitute the values 
of div d, rot h, div h and rot d taken from (17), (19), (18) and (20). 

4: (Page 18). The following considerations, showing, not only 
that the function (30) satisfies the differential equation (29) (which 
might be verified by direct differentiation), but also under what con- 
ditions it may be said to be the only solution, are taken from a 
paper by Kirchhoff on the theory of rays of light. ^) 

They are based on Green's theorem and on the proposition 
that, if r is the distance from a fixed point, and F an arbitrary 
function, the expression 

lias the property expressed by 

This follows at once from the formula 

^ 8r^ r 8r r dr^ ' 

which is true for any function of r, not explicitly containing the 
coordinates, and in virtue of which (6) assumes the form 

It is well known that 

rx = F[t+^) and rx = i^(^-y) 
are solutions of this equation. 



1) Ann. d. Phys. u. Chem. 18 (1883), p. 663. 



234 NOTES. 4 

Let 6 be the bounding surface of a space S throughout which 
■^ is subjected to the equation (29), P the point of S for which we 
want to determine the function, dS an element of volume situated 
at the distance r from F, S a. small spherical surface having P as 
centre, n and N the normals to s and 2, both drawn towards the 
outside. 

Introducing the auxiliary expression 



« = |-^(«+t). 



where i*" is a function to be specified further on, we shall consider 
the integral 

extended to the space between 6 and 21. 

In the first place we have by Green's theorem 

and in the second place, on account of (29) and (6'), 
Hence, combining the two results, 

+^i/(*lf-4?)''«- 

This equation must hold for all values of t. After being mul- 
tiplied by dt, it may therefore be integrated between arbitrary 
limits t^ and t^, giving 



t, 'h 



(?) 



From this equation we may draw the solution of our problem 
by means of a proper choice of the function F, which has thus far 
been left indeterminate. 



4 NOTES. 235 

We shall suppose that F{s) differs from zero only for values 
of s lying between and a certain positive quantity d, this latter 
being so smaU that we may neglect the change which any of the 
other quantities occurring in the problem undergoes during an interval 
of time equal to s. As to the function F itself, we shall suppose 
its values between £ = and £ = ^ to be so great that 

<) 
fF{s)ds = l. 



Since, for a fixed value of r, 



ti + - 



fF(f+{)dt=fF{8)de, 



it is clear that on the above assumptions 



and 



f-F(t+^)dt=.^^_^y 



(8) 



if we understand by x one of the functions of t with which we are 
concerned, and by t^ and ^j values of t, such tbat 

^i + f<0 and t,+~>S. 

It wiU presently be seen that, in the discussion of the equation (7), 
the formula (S) enables us to select as it were the values of ^ and to 
corresponding to definite moments. 

Let t^ have a fixed positive value and t.^ a negative one, so 

great that even for the points of 6 most distant from P, t^ -\ < 0. 

Then all values of i occurring in the last term of (7) are zero. So 

are also the values of -^ in that term. Indeed, 

at ' 



dt r 



F 



•('+y) 



and this vanishes for t = t^ and t = t^ because F'(a), Uke I (e) itself 
vanishes for all values of £ outside the interval (0, d). The last 
term on the right-hand side of (7) is thus seen to be zero. 



236 NOTES. 

The term containing cj may be written 

-fUsf.F{t+^)dt, 



where 

I o}FU+ ^)dt 
t, 
relates to a particular element of volume dS, at the distance r 
from P. Hence, on account of (cS) 

— j dt j %adS = — / ~^/ r\dS. 
By similar reasoning it is found that 

integ 
differential coefficient being equal to 

we have 



We have next to consider the integral containino; -t^- • This 
being equal to 

f,(|)4 + 7)+r.|-:^'('+T). 



'1 '1 



+ .. 



The first integral is 



fl iv) '^M(t + f ) dt ^Jl (I) rp^___ ,^ da, 
and the second expression may be integrated by parts: 

./'VfK*^'('+^)''"=/^£'''/*^'('+-0"' 

= fLtde\ri.FU + ^-f' ^^-d/f\%Fit + ^)dt 

J r on !^ V c/|^ J r dn J dt V c/ 

because both F\t^ + -) and FU^ + —J vanish. 



4 NOTES. 237 

Combining these results we find for the right-hand member 
of (7) 



-/ 



CO/ ,.\(IS 



We shall now suppose the radius R of the sphere 2] to diminish 
indefinitely. By this the first integral in our last expression is made 
to extend to within the immediate neighbourhood of the point P. 
The remaining terms remain unchanged, but for the quantity on the 
left-hand side of (7) we must take its limiting value for lim B = 0. 

As the integral over the sphere has the same form as that over 
the surface which we have just considered, we may write 






or, since the normal N has the direction of r, and since, at the 
sphere, r = jR, 



f 



1 /cip\ . 1 

1 



■a \a^/(.=--^) + 5^'''(,=-f) + cii'^(,=-f) ^'^^• 



Now, when R tends towards 0, the integrals with -^ vanish, so 
that the expression reduces to 

Let ip^ and j^j be the extreme values of ^/ _ jt\ on the surface 
of the sphere. Then (5) is included between 

in^fi and 4;rt^2-. 

But both Tp^ and ii^ have for their limit the value of ^ at the 
point P for the instant t = 0, say i^p^i^^^y so that the limit of (9) 
is seen to be 

and equation (7) ultimately becomes 



238 NOTES. 5 

This determines the Talue of ip at the chosen point P for the 
instant ^ = 0. We are, however, free in the choice of this instant, 
and therefore the formula may serve to calculate the value of Vp 
for any instant t; for this we have only to replace the values of ro, 

rp, -rA- and ip on the right-hand side by those relating to the time 

t Distinguishing these by square brackets, and omitting the 

index P, we find 



t- 



-lj¥''« + l^/|}[a-a4(f) W + il^Ml <*•• M 

The formula (30) given in the text is obtained by making the 
surface 6 recede on all sides to infinite distance, by which in many 
cases the surface integral is made to vanish. We may suppose, for 
example, that in distant regions of space, the function tp has been 

zero until some definite instant L. The time t to which the 

" c 

quantities [ip'\, Uj— L [i^] relate, always falls below t^ when r in- 
creases, so that, finally, all the quantities in square brackets become 0. 

5 (Page 19). When a vector A, whose components we shall 
suppose to be continuous fimctions of the coordinates (cf. § 7) is 
solenoidally distributed, so that 

div A = 0, {11) 

we can always find a second vector B such that 

A = rot B. 

It suffices for this purpose to put 

B^ 1 /-rotA^^ 
47t^ r 

Indeed, we find from this, if we use equation (2) of Note 1 and 
the above equation (11), that 

^■°t B = i r^'^ ds=- -i- f^ds, 

and this is equal to A in virtue of Poisson's theorem. 



» NOTES. 239 

In this demonstration we have used the theorem that, if (n is 
continuous, a potential function of the form 



/" 



dS 



may be differentiated with respect to one of the coordinates by simply 
differentiating m under the sign of integration with respect to the 
corresponding coordinate of the element dS. 

Now, equation (18) shows that the magnetic force h is solenoi- 
dally distributed. Therefore we can always find a vector a such that 

h = rot a. (12) 

After having done so, we may write for the equation (20) 

rot(d + |a) = 0, 
showing that the vector 

d + - a 

c 
must be the gradient of some scalar function — tp, so that 

d = -^ a-grad<)r). (13) 

It must be observed, however, that the vector a and the scalar 
function qp are left indeterminate to a certain extent by what precedes 
(though in each special case h and d have determinate values). 
Understanding by ap and cpg special values, we may represent other 
values that may as well be chosen by 

a = ao - grad X, cp = q,^ -\- --;{;, 

where x is some scalar function. We shall determine it by sub- 
jecting a and (p to the condition 

diva = -^9, (14) 

which can always be fulfilled because it leads to the equation 

A;);--^i(; = divao + -9'o; 

which can be satisfied by a proper choice of x- 

The differential equations (31) and (32) follow immediately 
from (17) and (19), if in these one substitutes the values (13) and (12). 
Indeed, (17) assumes the form 

--diva-A(p^Q, 
i. e., in virtue of (14) 

^T — -^9 = — Q, 



240 NOTES. 6, 7 

and (19) becomes 

rot rot a = s- a grad m -] p V, 

or (cf. Note 1) 

grad diva — Aa = 5- a grad^; + — pv, 

for which, on account of (14), we may write 

1 .. 1 
Aa va = pv. 

6 (Page 20). Our solution is not a general one because we 
have made the assumption that the surface integral in (10), Note 4 
vanishes when the surface 6 recedes to infinite distance. It is to be 
observed, however, that any other solution may be put in the form 

where ip' is some function satisfying the equation 

In the terms of the physical problem with which we are con- 
cerned, we may say that the electromagnetic field determined by 
(33) — (36), (which may be considered as produced by the electrons), 
is not the only one that can exist; we can always add a field satis- 
fying at aD. points of space the equations (2) — (5) for the free ether. 
Additional terms of this kind are excluded by the assumption made 
in the text. 

Of course, a state of things for which the formulae (2) — (5) 
hold, can exist in a limited part of space; the beam of plane polarized 
light represented by the equations (7) is a proper example. Such a 
beam must however be considered as having its origin in the vibra- 
tions of distant electrons, and it is clear that, if we wish to include 
the source of light, we must have recourse to equations similar to 
(33)-(36). 

7 (Page 21). Let the centre of the electron move along the 
axis of X. Then it is clear that a^^ = 0, a^ = 0, and that (p and a^ 
may be regarded as functions of t, x and the distance r from the 
origin of coordinates. Indeed, (p and a^ must be constant along a 
circle having OX for its axis. 

Putting 

(p=fi{t,r,x), a^ = f^{t,r,x), 



s 




NOTES. 




one finds 










-. = -^. 


dx c dt 


dfr X df, 
dx r dr 




"> — 1? = 


r dr> ^ 


dtp z d 
dz ~ r d 



241 



Hence, d may be considered as the resultant of two vectors, one 

having the direction of OX and the magnitude 4A — '^^' , and 

° c dt dx' 

the other the direction of r and the magnitude — -J^ ■ 
The components of the magnetic force are 



h - ^^^ - 

" dy 


d&y 

dz - 


= 0, 


"y dz 


dx 


z df, 
r dr ' 


^ ex 


_ 3 a,,- ^ 
dy 


y df, 
r dr 



so that h is at right angles both to OX and to the line r. 

What is said in the text about the electric and the magnetic 
lines of force foUows immediately from these results. 

8 (Page 22). In establishing the equation of energy we shall 
start from the formula (23). For an element of time dt the work 
of the force exerted by the ether on an element dS oi the chai-ge 
is represented by the scalar product of the force fgdS and the path 
\dt. Hence, the integral 

A=^fQ{f-\)dS 

represents the total work done by the ether per unit of time, but 
this work depends entirely on the first part of the vector (23), since 

the second part — [V • h] is perpendicular to the velocity V. Consequently 

A =/()(f • 'j)dS =/?(d • yi)dS =/(d ■Q\)dS, 
and, if the value of ^V is taken from the equation (19), 

A = cl{d-rot\\)dS -f(A-A)dS. {15) 

Written in full, and with the terms rearranged, the first integi-al is 

./(Kll'-".§)+(''.§-''.t)+Kt--.t)l"«'W 

and here each term may be integrated by parts. Thus, denoting by 



Lorentz, Theory of electrons. 



16 



242 



NOTES. 



a, /3, y the angles between the normal n to the surface and the 
positive axes, 






where [d • h]^ means the first component of the vector product [d ■ h]. 
If the remaining parts of (16) are treated in a similar way, 
the first integral in (lo) becomes 



y(d-roth)d<S = 

= —J { [d • h]^ cos « + [d • h]j, cos /3 + [d • h^ cos y ] de 

+/(h ■ rot d) dS = - /[d • \i]j6 +f{\\ ■ rot d) dS. 



(17) 



The formula (37) is now easily obtained if it is taken into 
account: 

1° that, in virtue of (20), the last term of (17) may be re- 
placed by 

-^J(b.b)dS; 
2° that 

(dd) 



2 dt ' *-' ^ 2 dt 



We may notice in passing that the equation (17) expresses a 
general theorem. Denoting by A and B any two vectors, and by a 
the bounding surface of a space S, we always have 

j^A • rot B)dS = -J [A • R]j0 +f(B ■ rot l\)dS. 

9 (Page 26). The deduction of the formulae for F is much like 
that of the equation of energy. Instead of (43) we may write 



=/{?d+|[?v-h])cZS, 



and here, in virtue of (17) and (19), me may replace q by divd, 
and pV by c rot h — d. Hence 



= Adivd- 



d + [rot h ■ h] - i [d • h] I dS 



S+t+©-.+(^--S)''.-(g-t)''.H* 



10 NOTES. 243 

But 

[d.h] = |^[d-h]-[d.h] = A[d.h]-c[rotd.d], 

so that, if we determine the part F^ of the resultant force by the 
formula 

the remaining part is given by 

Fi =f[ div d • d + [rot h • h] + [rot d • d] ) ^5- . 

Leaving aside for a moment the term depending on the magnetic 
force, we have for the first component of Fj 

/!( 
=/( 

=J |y(^/- V-^/) <=os« + d,d^ cos/3 + d,d, cosy) d0 

=^J ~\2dJ^-A^co8cc]d6. 

The part of Fj that depends on h leads to a result of the same 
form, the reason being that Fj becomes symmetrical in d and h when 
we add the term div h-h, which is zero on account of (18). 

10 (Page 29). The stress on a surface element of any direction 
and situated anywhere in the space considered can be calculated by 
means of the formulae (48); if one takes the mean values for a long 
lapse of time, it will be found to be at right angles to the element. 
In other terms, there is a normal pressure whose magnitude is given by 

i> = i { (d/) + (d/) - (tJ.^) } + i { (h/) + (h/) - (h/) } , (18) 

if we lay the axis of x normally to the element, and denote by (d^^), 
etc. the mean values in question. 

We shall now apply to two particular cases the result found in 
§ 19. In the first place, we may take for 6 a closed surface wholly 
lying within the envelop. Then (cf § 20, b), since F = and, in the 
mean, Fg = 0, the pressures p acting on the surface must destroy 
each other. This requires that p be constant all through the ether. 

Next, considering a flat cylindrical box that contains an element 
of the wall (cf. Fig. 1, p. 28 !, we can show that the pressure p reaUy 
may be said to be the force exerted on the walls. 

16* 



244 NOTES. 11, 12 

The pressure }'> having the same intensity at all jaoints, we may 
as well replace it by the mean of the values which, for determinate 
directions of OX, OY, OZ, the expression {18) has at different pla- 
ces. Hence, if mean values of this kind are denoted by a horizontal bar 

p = ^{(57) + (O - (57)} + i{(V) + (1^) - ^K')]- 

But it is easily seen that the order of the two operations of 
taking the mean — one relating to time and the other to space — 
may be inverted, and that in the stationary state which we are con- 
sidering the mean values indicated by d^^ etc. are independent of the 
time, so that, after having calculated them, it is no longer necessary 
to take their time-averages. Our formula therefore takes the form 

J, = i(d7 + d7- ^}) + -i-(n7 + h/"- h/) . 

11 (Page 30). The formula (51) is obtained if, in the trans- 
formations given in Note 9, we omit all terms containing q. We 
may, however, also proceed as follows. 

The resultant force in the direction of x, so far as it is due to 
the electric field, is given by the surface integral 



^/' 



[2d,d„-d^cos«}c?<?, 

for which we may write (see the end of Note 9) the first component of 

j'{divd-d + [rotd-d])c/6', 

and to which we must add a similar expression depending on the 
magnetic field. Hence, since div h = 0, and, on the assumption now 
made, div d = 0, 

Fi =J'{[roth-h] -f [rot d • d] } £^5', 

or, if we use equations (4) and (5), 

F, = |^/{[d.hl-[h.d]}</,S'=lJ{[d.h] + [d-hl}cZ-S' 

12 (Page 32). Let u, v, w be the components of the velocity 
of the ether at the point {x, y, z) and the time t. Then, by a well 
known theorem, the acceleration in the direction of x is given by 

3m I 3m , da , du 
ot dx ' dy ds ' 



13, 14 NOTES. 245 

so that, if fi is the density, and X the force acting on the element 
dS in the direction of x, we have 

When u, v, w are very small, we may neglect the terms tt-„— etc., 
and add the term u^^, which is likewise of the second order of 

c t 

magnitude, because in the case of slow motions, the change of the 
density per unit of time is very small. It follows that 

the mathematical expression for the statement made in the text. 

13 (Page 35). The value 95 of the scalar potential that exists 
at the time t at the point (x, y, 0) of the ether, will be found at 
the time t + dt at a point whose coordinates are x + wdt, y, z. As 
the value of the potential for these new values of the independent 
variables may be represented by 



we have 



, Sep J, d(p J, 



dt ' ox 



dw dw 

-1^ = — w^ 
ct ox 



Applying the same reasoning to the function -^ , one finds 

dt^ ~ dx \dt) ox' 

14 (Page 36). Let S' be a system without translation, and let 
two points, the one in the moving system S, with the coordinates 
x, y, z, and the other in S' with the coordinates x\ y, z — the re- 
lation between x and x being as shown in (58) — be said to cor- 
respond to each other. Then corresponding elements of volume, 
dS and dS', are to each other in the same ratio as x and x , so that 

d8'^(\-f)-^i^dS, 

and if they are to have equal charges, the density p' in dS must be 
related to the density q in dS as follows: 

p'=(i-^T^p- 



246 NOTES. 14 

Pois son's equation, whicli determines the scalar potential qp' in 
the stationary system may therefore be written in the form 

showing, on comparison with (59), that at corresponding points 

q,' = (l- f)'l\, g) = (l-pyy^q)'. (19) 

The quaatities relating to the moving system S may now be 
expressed in terms of those that belong to S'. 

In the first place we have, on account of (58 j and {19), 

|?^=(i_/32-i|jp; |^_n_^.)-i/2|?: |^=(i_^^)-i/2^. 

Further, by (33) and (34), since 

K-H, a^ = 0, a. = 0, 

^ ox ^ ox' 

^ c ^ dx ^ ^ ox' 

"y dy' ' dz' 

"^ "' "v cz Pdz' "- ay Pdy 

The electric energy is therefore given by 

^=:/|ci-«'6l)'+(lfr+(M)V« 

= i/( (1 - /i7» (II)' + (1 - W-" [(i|)" + fsl)"] 1 iS, 

and the magnetic energy by 

Finally, we have for the components of the flow of energy 

s.=.(d,h.-dA)=o^(i-^rM(iir+(i?ri' 

s. = c{Ax, - ^K) = - c/? (1 - n ''■' g % , 



m 



{21) 



15 NOTES, 247 

and for those of the electromagnetic momentum 

G,= i«l-rt-"'/|(g)'+(|f)Vs', m 

15 (Page 37). A charge uniformly distributed over the surface 
of a sphere may be considered as the limitLag case of a charge distri- 
buted with uniform volume density over an infinitely thin spherical 
shell having the same thickness at all points. When the moving 
system S is of this kind, the stationary system S' of which we have 
spoken in the preceding Note, is an elongated ellipsoid of revolution 
whose semi-axis a and equatorial radius h are equal to 

a = (l-/32)-i/2E, l^B, {23) 

and which carries a charge uniformly distributed through an infinitely 
thin shell bounded by the ellipsoid itself and another that is similar 
to it and similarly placed with respect to the centre. The total 
charge must be taken equal to e, the charge of the sphere, because 
corresponding elements of volume in S and S' have been supposed to 
carry equal charges. 

Let the centre of the ellipsoid be chosen as origin of coordina- 
tes, OX! being placed along the axis of revolution, and let x, y, 
be the coordinates of an external point P. If we understand by X 
the positive root of the equation 

- -^l!±^ = l, (24) 



where 



p2 = a2 _ 2,2^ 
the potential at P is equal to 

w = - — log V- ' — • 

It is to be noticed that, for a given value of I, the equation 
(24) represents an ellipsoid of revolution confocal with the given one; 
therefore, the equipotential surfaces are ellipsoids of this kind. The 
charged surface itself is characterized by the value X = ¥, and I in- 
creases from this value to '>o as we pass outwards. The potential 
is equal to 

<Pn = i, — log — ■ — 
^0 8 7tp °a—p 



248 



NOTES. 



15 



at the charged surface, and has the same Talue at all internal points. 
The integrals to which we have been led in the preceding Note need 
therefore only be extended to the outside space. 

In effecting the necessary calculations we shall avail ourselves of 
the theorem that the integral 



l,/'((feT+(lf)"+©)V*' 



is equal to the electric energy -lecp^' of the charged ellipsoid. Hence, 
putting 

^. =/©■«■. ^.=/i(itr+(ii)>^-. 

we have 

J^ + J^='--log^- + P.- (25) 

In order to find the integral J^, we shall divide the plane X'OF 
into infinitely small parts by the series of ellipses 






= 1, 



and the system of hypei'bolae 



^ y__ = 1 



[26) 



{27) 



where jn ranges from to p^. Confining ourselves to the part of the 
plane where x and y are positive, we have for the coordinates of 
the point of intersection of {26) and (27) 



^' = ^V{p' + ^)i.p'-^), 



y 



rV^f^, 



(28) 



and for the area of the element bounded by the ellipses A, I -\- dX 
and the hyperbolae fi, ft + f^f- 



d6 = 



8x' 


dx 


dl ' 


dii 


dy 
dl' 


dy 



dldfi = 



1 X-\-ti, 



d X d[i . 



We shall now take for dS' in our integral the annular element 
that is generated by the revolution of this plane element around OX', 
so that 



dS' = 2%yde -■ 



■nil + n) 



Since cp' depends on X only, we have 



dXd^i. 



Of) 

dx 



dcp dl 
dl dx' ' 



15 NOTES. 249 

and here the last factor has, in all parts of the ring, the value de- 
duced from {26) for a constant y, 

n_ ^ 'ix\p^-\-i)x' 

dx l^x'^+{p^-{-l)\f' 
or, in virtue of (28), 



ii'=wiwpy^p'+'')^p'-^)- 



It follows from these results that, in order to find J^, we must 
integrate the expression 



p\X + iC} 



V{p'+X){p'-i^)(-g)dXdi,. 



If we take and p^ as the limits of ,ii, V and oo as those of X, 
we shall find the part of J^ that is due to the field on the positive 
side of the j/^^-plane; we must, therefore, multiply the result by 2. 

Since 



P 



flgEf ,1^ = _ 2p + y^+l logl^lA-ti' , 

u 

and 

\dl) ~ 6i^n^{p' + X)' 
the final result is 



T == — 
1 16jrp 



J \ ^Vp' + x-p Vp'' + ^> 



The indefinite integral is 

Vp' + ^-p 

and since this vanishes for A = ^, and is equal to 

bnos'^^-2a27 
° a — p 

for I = V, the integral J^ has the value 

In our present problem the values of a and h are given by {23), 
so that 



250 NOTES. 16, 17 



7 = « 

"^2 l63tJB|3' 



(1 _^2)i/2[_ 2^ + (1 + /30 logrzr|] ■ 



Substituting these values in the formulae (20), {21) and (22), 
we get the equations (61), (62) and (63). 

16 (Page 38). The electromagnetic momentum G and the velo- 
city W having the same direction, we may write 

G = aw, 

where a is the ratio between their magnitudes | G , and [ W j . It is a 
function of | W | . 

Differentiating with respect to t, we find 

'' dJ ~ ^ "^ Ji ~ U ~dt ~ J]w! ^dT 

But 

so that 

= -«(j +J )-|w|dl^J =-dTw! ^"'^IJJ ""''J 
(^|G| ., IGj .„ ,., „.„ 

17 (Page 48). Take the simple case of an infinitely long cir- 
cular metallic cylinder of radius a^, surrounded by a coaxial tube 
whose inner radius is a^. When a current i is passed along the core 
and returned through the tube, the magnetic energy, so far as it is 
contained in the space between the two conductors, is equal to 

" log^ 



per unit of length; this expi-ession is of the order of magnitude 



4jtc^ 



{29) 



a, 



when - is some moderate number, 
o, 

On the other hand, if, per unit of length, the two conductors 
contain iV^ and N^ electrons, moving with the velocities v^ and v^, 
the sum of the amounts of energy that would correspond to the mo- 
tion of each of them is 



18 NOTES. 251 

if we suppose the mass of the corpuscles to be wholly electromagnetic. 
The current being 

we may write for our last expression 



12 



-^EAi;+i-)- (•^^) 



The experiments on self-induction have never shown an effect 
that may not be accounted for by the ordinary formulae for this 
phenomenon. Therefore, in ordinary cases, the value of {30) must be 
much smaller than that of (39), from which it may be inferred that 
iVji? and N^R are great numbers. 

18 (Page 49). In the following proof of the formula (76) we 
shall confine ourselves to an electron having a rectilinear translation 
parallel to OX. with variable velocity v. Let (J be a definite point 
of this electron and P a point of the ether, within the space occu- 
pied by the particle at the time t for which we wish to calculate 
the force. Let x', y', z be the coordinates of P, and x, y, z those of 
the point Q at the time t. 

Among the successive positions of Q there is one Q^ such that 
an action proceeding from it the moment it is reached, and travelling 
onward with the speed of light c, will arrive at the point P at the 
time t. If we denote hjt — r the time at which this „effective" posi- 
tion, so we may call it, is reached, we have for the coordinates of Q^ 

x^ = X — vx -\- -^-i»r^ — Yvx^ -\- ■ ■ ■ {31) 

Ve = 2/> ^e= ^, 

and, since Q^P must be equal to cr , 

{x, -xj + {y- yj -K {z, - zj = cH\ {32) 

By means of these relations x^ and x may be expressed in terms 
of X, y, z. Putting QP = r, so that 

r- = {x - x'Y + (2/ - y'f + (^ - ^'Y, 

and considering v, c, V, ... as so small that terms of the second order 
with respect to these quantities may be neglected, we may substi- 
tute in {3t) T = — , by which we find 



252 NOTES. 18 

Substituting this value in {32), we get 

r = J: _ ^(^_^') + ^(^_^')^ _ JL(i,_a;')r2---- 
c c^ ^ •' 2c' ^ ' fac ^ 

It follows from (33) that the points Q which, at the time t, 
are situated in an element dxdyds, have their effective positions in 
an element dx^ dy ds, where 

V X — X ^^ V , ,^ 'V 



dx^^\l-^"^^+:.{x-x')-,-Ax-x')r+--\dx. 



Hence, to each element dS of the electron taken in the position 
which it has at the time t, there corresponds an element of space 

dS^=[l-^^^-^''~ + ^.{x-x')-^]^,{x-x)r^--}dS, 

in which, at the time t — r, there was a density q equal to that 
existing at the time t in the element dS, this charge having a velocity 

V — it -\- \'vx'^ — •■■ , 

or, with a sufficient degree of approximation 

The distance of the element dS^ from the point P is given by 
f ^ V X — X , b , ,, v , '\ i 1 

d ^ 

SO that the quotient — in the equation (35) must be replaced by 



-^ =[1 + j^,{x -X) - —,{x-x)r + -..j-. 



The factor here enclosed in square brackets may be omitted in 
the formula for the first component of the vector potential; here, 
however, we must replace v by the expression (34). In this way 
we find 



1 r / i ,«, \ds 

1 — /PU' >' + ir-i»"" — , 

^ncj ^ \ c ' 2c^ / r ' 



the integrations being extended to the space occupied by the electron 
at the time t. 

We shall now proceed to calculate the electric force f at the 
point P. It may be observed in the first place that we need not con- 



IS NOTES. 253 

sicler the term — [v • h] in (23), because the magnetic force h itself 

is proportional to v. Hence, by (33), the first component of f, to 
which we may limit ourselves, is equal to 



f = - ^'P 



dx c " 



As the differentiations may be effected under the sign of inte- 
gration, we have 

and, since / QdS = e, 

t - i-J^"^'' - sij^ It + ^-^1 ^^+^- (^^) 

In order to find the resultant force we must multiply this by 
q'cIS', where dS' is an element of volume at the point P, and ^' the 
density at this point; we have next to integrate with respect to dS' 
From the first term in (55) we find 0, and from the last term 



6 5rc' 



agreeing with the expression (76); these results are independent of 
the shape of the electron and the distribution of its charge. As to 
the middle term in (55), it leads to the force 



.',./<■'"•/<- (v + ^.^>« 



In the case of a spherical electron the charge of which is distri- 
buted symmetrically around the centre, we may write ^r^ instead of 
{x — x'Y, so that we get 

Now, if the charge lies on the surface, the integral / ydS has the 

value p- at aU the points where the density q' is different from zero. 
Therefore (36) becomes 



254 NOTES. 19 



^jQ'd8' = 






in accordance with the result expressed in (72). 

What has been said in § 37 about the representation of the re- 
sultant force by a series, each term of which is of the order of 

magnitude — in comparison with the preceding one, is also confirmed 

by the above calculations. 

19 (Page 50). Let us fix our attention on the effective posi- 
tion M (cf Note 18) of a determinate point of the electron, for in- 
stance of its centre. If this position is reached at the time t^, pre- 
vious to the time t for which we wish to calculate the potentials at 
the distant point P, and if the distance M P is denoted by r, we have 

r = c(t-t,). (37) 

Choosing M as origin of coordinates, we shall understand by 
Xjp, 2/p, Sp the coordinates of P. 

Let us further seek the effective position (x^, y^, z^ of a point 
of the electron whose coordinates at the time t^ are x, y, z. This 
effective position M' will be reached at a time ^^, a little different 
from ^g; if we put 

the interval r will be very small. The coordinates x, y, z are so 
likewise, and a sufficient approximation is obtained if, in our next 
formulae, we neglect all terms that are of the second order with 
respect to these four quantities. 

The condition that M' be the effective position of the point con- 
sidered is expressed by 

M'P = c{t-Q = c{t-t^~x). {38) 

But, if V is the velocity of the electron at the time #(,, we may 
write for the coordinates of ilf 

Xe-x + v^T, y, = y + y,jX, z^ = z-\- v^t, {39) 

so that {38) becomes 

{Xp-X~ S^xf + {y^-y- s^rf + (^^ _ ^ _ v,t)^ = cX< - <o - ^f, 

or, on account of {37), and because 

r 



20 NOTES. 255 

is the component V^ of V along the line MP, 

2(XpX + ypy + ZpZ) + 2v,.rT = 2crt, 
giving 

The points of the electron which, at the time t^, lie in an ele- 
ment dS, have their effective positions in an element of space dS^, 
whose magnitude is equal to the product of dS by the functional 
determinant of the quantities {39) with respect to x, y, z. The value 
of this determinant is 



or, in virtue of (40), 



1 + V ^- + V ^^- + \~ 






As to the distance r in the denominators of (35) and (36), we 
may take for it the length of MF, and in the latter of the two for- 
mulae we may understand by V the velocity of the electron at the 
instant t^^. In this way the general equations take the form 

fp = -, — f odS^, a = -7 I odS^, 

which is equivalent to (79) because (p being equal to the density 
existing at the time t^ in the element dS) 



fodS^ -^ fo dS = — ~ • 



20 (Page 51). As the field depends on the differential coefficients 
of the potentials, we have first to determine these. In doing so, we 
shall denote by x, y, z the coordinates of the distant point P for 
which we want to know d and h. 

If we change by dt the time t for which we seek 90 and a, 
keeping x, y, z constant, it will no longer be the same position of 
the electron which is to be called the effective one. Besides, the new 
effective position will be reached at a time slightly differing from t^ 
and will lie at a distance from P different from r, the changes being 
connected with each other by the formula 

dr = — S^dtf^, 

where v^ has the meaning explained in § 38. 
Differentiating equation (.57), we find 

— \^dt^ = c{dt-dQ, 

dL = dt. 

* c— Vr 



256 NOTES. 20 

It appears from this that, by the change now considered, the 
value of some quantity i' corresponding to the time t^, is altered by 

so that we may write 

cl[ii>] _ c r8i>i 

the square brackets always having the meaning formerly assigned 
to them. 

In applying this to the expressions (79), we shall suppose the 
distance r = MP to be so much greater than the dimensions of the 
electron that, in the final formulae for d and h, we may neglect all 

terms of the order ^ . Doing so we may treat as constants the 

three cosines in the equation 

V^ = v^. cos (r, x) + Vj, cos {r, y) + v^ cos {r, z); 

indeed, their differential coefficients are of the order — , and in (p there 
is already a factor — . Consequently, 

Tt = J- ^°^ ^^' *) + Jy *^°^ ^^' y^ + J^ ^^^ *^^' ^^ ^ ^'■' 
and since the factor — in qo may be considered as constant 
d<f ^ e_ d [V;.] ^ ^ |- j 1 

*' ° '"['(' -yn ""^*«H'-7')'] 

If, finally, we neglect all terms that are of the second order 
with respect to the velocity and the acceleration of the electron, we 
have the further simplification 

Similarly one finds from the second of the formulae (7U) 



dt iTtcr 



[]]■ 



We have next to calculate the differential coefficients with respect 
to the coordinates. Consider first an infinitely small ' displacement of 
P in a direction /i at right angles to MF. The distance MF not 
being altered by this, and t being kept constant, neither the instant t^ 
nor the efPective position M are changed. As we may again leave 



21 NOTES. 257 

out of account the change in the direction of r, we conclude that 

^ = 0, |? = 0. 

oh ' dh 

The differential coefficients with respect to the direction of r 
are easily found by the following device. If P is displaced over a 
distance dr along MF prolonged, t being increased at the same time 

dr 
by dt = — , the effective position of the electron and the time t^ re- 
main unaltered, so that, since the denominator r need not be diffe- 
rentiated, 

8qp 1 , Sqp dr „ dtp I d<f 2* _ 1 3a 

dr ^ ^'di'c ~ ' Jr c 3< - Jr c di' 

Combining this with the former result, we find for any di- 
rection Tx, in the case both of the scalar and of the vector potential, 

and particularly 

d /■ \ S d / \ S d /■ \ S 

^— = cosir, aj)^— , 5— = cos (r, v) 3- , ^- = cos (r, «) 5— • 
8x ^' ■' dr ' dy ^ ' "■' dr' dz ^' 'dr 

Using these relations one will find without difficulty the for- 
mulae (80) and (81). 

21 (Page 51). In the formulae (80) each component of d is 
represented as the difference of two terms. The terms with the 
negative sign may be considered as the components of the vector 

« -i 

inc^r ' 

and the terms with the positive sign as those of the vector 

where we have used the parentheses in order to indicate that the 
component j,. is here itself regarded as a vector. Understanding (j^) 
in a similar sense, so that 

J = (\r) + Op), 

we have 

h = — V[j-k] = -[dl<]- 
The magnetic force is therefore perpendicular both to d and 

Lorentz, Theory of electrons. 1' 



258 NOTES. 22 

to k, and its direction is such that the flow of energy c [d • h] has 
the direction of k, away from the electron. The intensity of the 
flow is c|d| |h| = cd^- 

22 (Page 53). As an interesting application of the formula 
found for the resistance, we shall calculate the damping of the 
vibrations of an electron. Suppose the particle to he subjected to 
an elastic force — /q , where q is the displacement from the position 
of equilibrium, and f a positive constant. The motion in the direc- 
tion of OX is determined by the equation 

a particular solution of which is found by taking for q^ the real 

part of 

pttt 

where s is the basis of natural logarithms, and « a complex constant 
determined by the condition 

ma' = -f+ -'^-,aK (41) 

If the last term has but a small influence, we may replace in 
it cc by the value given by the equation 



Hence, putting 
we have 



ma^ = — f. 

m ' 

a = m — 



and, introducing two constants a and ^j. 



q^ = «e i2«mc» COS (nt+p). 
This formula shows that in a time equal to 

12 7cmc^ 



the amplitude falls to — of its original value. 



Taking for m the value (72), and writing T for the time of 



in 



vibration — , 2 for the wave-length, we find 



"^ ~ 2%^Ii'^- 



23 NOTES. 259 

If we substitute for B the value given in § 35 we have for 
yellow Ught (;i = 0,00006 cm) 

r = 2 • 10' T, 

showing that the damping would be very feeble, and that we have 
been right in supposing the last term in (41) to be very small. 

This question of the damping of the vibrations is important be- 
cause, the slower the damping, the more wiU the radiation present 
the character of truly homogeneous light. We can form an opinion 
of the degree of homogeneousness by making experiments on the 
visibility of interference fringes for various values of the difference 
of phase; in fact, when this difference is continually increased, the 
fringes can remain clearly visible for a long time only if the light is 
fairly homogeneous. A small degree of damping is thus found to be 
conducive to a good visibility of the fringes, a conclusion that is 
readily understood if one considers that the interference becomes in- 
distinct when the intensities of the two rays are very different. This 
must be the case whenever the vibrations in the source have con- 
siderably diminished in amplitude between the instants at which the 
interfering rays have been emitted. 

The result of the above calculation is in satisfactory agreement 
with the experiments of Lummer and Gehrcke in which, under fa- 
vourable conditions, interferences up to a phase difference of two 
millions of periods were observed. 

23 (Page 56). In each successive differentiation with respect to 
one of the coordinates, of the expression found for —^\'we have to 
differentiate both the goniometric function and the factor preceding it. 
These operations introduce factors of the order of magnitude — = -^^ 

(if A denotes the wave-length) and — Consequently, in as much 
as r is very much greater than A, we may confine ourselves to the 
differentiation of the goniometric function. 
Thus, for example, 

a =_^^[PJ = _-!L^-sin|«(^-~M4-i.|, 

d =_^_i|^^=-^i-(_-;+i)cos{^(<-^)+i.|- 

It is easily verified by means of the expressions (95) that d 
and h are at right angles both to each other and to the line r, and 

17* 



260 NOTES. 24 

that they have equal amplitudes. The formulae represent a system 
of plane polarized waves, whose amplitude changes in the inverse 
ratio of the distance r as we pass along a straight line drawn from 

the radiating particle. The flow of energy changes as -^j ■ 

24 (Page 58). Considering any one of the dependent variables, 
say \\), first as a function of x, y, z, t, and then as a function of 
x, y', s , t', we have the following relations, arising from (96) com- 
bined with 

t'=t-^^{\N^x + \N,^y + \N^z), 

as we may write instead of (97) if the square of — is neglected, 

ci/) Si/) dx . dtp 8y' _, dip dz dip dt' dip w^, dip 

dx dx' dx dy dx dz dx dt' dx dx' c^ St' ' 

dip dip v/,j dip dip dip Wj dip 

dy dy' c^ dt' ' dz dz' c^ dt' ' 

dt dt' ^ dx V dy ' dz' ' 

By this the equation (17) becomes 



c ' dy dz c^ L ^ ct v dt ' dt J ^ 



dx' ' dy 



In the terms multiplied by W^, W , W^ we need not distinguish 
between the differential coefficients with respect to t\ x' , y , z , and 
those with respect to t, x, y, z. Hence, in virtue of (19), we write 
for the terms enclosed in square brackets 

'""A^'-dPi + ^*v(g7- g^') + '"^A-di'-jy) - (^•^)?- 

In the last term V may be replaced by U, because we are con- 
stantly neglecting the square of W, and we are led at once to equa- 
tion (100) if we keep in mind that 

tl. + |(w,h_.-w,h,; = d;, etc. 

Let us next transform the first of the three equations taken 
together in (19), namely 

dy dz e \^^^^ dt) 
It assumes the form 



25 NOTES. 261 

or, if pW^ is replaced by ' 

/ad an, adA 

^^Kdx'^ dy'^ dz'j' 
and if the terms are arranged in a different order, 

|^(h.-4(w.d,-w^dj)-^(h^-l(wA-w,dJj 

This is the first of the equations contained in (102). 

26 (Page 59). We shall begin by observing that the potentials 
(p and a' satisfy the differential equations 

A<p'-^-^^ = -9, {42) 

g2 ^2 gS 

(cf. Note 4), where A is now an abbreviation for ^-^j + ^-ij + o— f^ , 
and that they are mutually connected in the following manner: 

di-^'=-||7' + ^V(w-a'). {M) 

In order to prove this latter formula we shall start from equa- 
tion (5) of Note 2, which, in terms of the new variables, may be 
written 

|^_w,^,-w,f^.-w,|% + div(9V)--V(w.« = 0, 

or, if the square of W is again neglected, 

|f + d,v(,u)-J,(w.^) = 0. (.5) 

If, in an integral of the form (104) or (105), the factor by 

which --_- is multiplied is a continuous function of the local time t' 

and the coordinates x', y, z of the element dS, the partial derivatives 
of the integral with respect to t' or to the coordinates of the point 



262 NOTES. 25 

for whicli it is calculated, are found by simply difPerentiating the 
said factor with, respect to t', or x, y, s , the differential coefficient 

being again taken for the value H ■ of the local time. 

According to this rule 

dt' 



intcj r L 8t _i ' 



from which we infer that 



(»--)=.:o/i[(»-^)]^«- 



In virtue of (45) these values verify the equation (44) and it is 
further found by direct substitution that the fundamental equations 
(100)— (103) are satisfied by (106) and (107) (see, however, Note 6). 
We have, for example, 

divd'= — — diva'— A^'+ — A(wa'). 

But by (44) 

diva=--g-?^ + -^(w.a), 

so that the foregoing equation assumes the form 

divd'= j, ^^ - Acp'-^iw-a') + iA(w.a'). 

The two terms containing a' are equal to 

i(w.(Aa'-ia'|). 

and in virtue of (42) and (43), the right-hand side of the equation 
becomes identical with that of (100). 

No difficulty will be found in the verification of (101) and (103). 
As to equation (102), we find from (107) (cf Note 1) 

rot h' = rot rot a' = grad div a' — A a', 

and, if we use (44), (43) and (106), 

rot h' = - ~- grad q>' + V grad (w . a') + -% U - ^ a ' = ^ (d' + p u). 



26 NOTES. 263 

26 (Page 59). The problem may be reduced to that of deter- 
mining the field due to a single moviug electron (cf. §§ 38, 41, 42 
and Note 19). Let P be the distant point for -which we want to 
calculate the potentials qp' and a' at the local time t', and M a de- 
finite point of the electron, say its centre, in its effective position, 
so that, if tf,' is the time (local time of M) at which it is reached, 
and *• the length of MP, 

r==c{t'-t,'). {46) 

Choosing M as origin we shall call x'^,, y'p, z'^ the coordinates 
of P, x, y, z those of some point Q of the electron at the time t^ 
(local time of Jf ), x^, <//, z^ the coordinates of the effective position 
Q^ of this point, and f^' + t (local time of M) the time at which it 
is reached, so that, according to (97), the local time of Q^ itself is 
then represented by 

K = C + 1 - ^ (w,..; + w^2/; + w,<). 

The condition that Q^ be the effective position of the point con- 
sidered is expressed by . an equation similar to (46), namely 

$,P = c(r-0, 

or, taking the square on both sides, 

{xp - x',f + {y'p - y;f + {zp - z',f 

= c\t'-t^-xf + 2(^'-V-*)(wX + w,2/; + w,,?;). 

The interval t being very short, we may write 

<=a;'+u^r, y',=y'\\iyt, z;=z'+U^r, 

by which, if terms of the second order with respect to x, y', z', r 
are neglected, and if (46) is used, our condition becomes 

— (x'pX + y'py' + z'pz) — ru^t 



= - ret + - (w^x + W^y' + w,/) + -- (W • u)t, 
(scTpx' + y'py' + z'p/) + y (w^.r' + w^y' + w./) 



r(c — U^) (w- U) 



(47) 



Here u^ means the component of u in the direction of MP, the 
product rU,. having replaced the expression x'pU^ + y'p\i,j + z'pU,. 

Having got thus far we can again distinguish between an ele- 
ment dS of the electron in its position at the instant t^' (local time 



264 NOTES. 26 

of M) and tlie element dS^ which contains the effective positions of 
the different points of dS, the ratio between the magnitudes of these 
elements being given by the functional determinant of xj, yj, z' 
with respect to x', y, /, i. e. by 

w , dt , dt , dr 
' ^ 3a; ' y dy 'dz 

We shall retain only the terms of the first order with respect 
to U^, Uj,, u,. Doing so, we may neglect in g^'ig— m g/ ^^^ terms 
containing these velocities, so that (47) gives for the determinant 



c 



Finally we have the following equations, similar to those which 
we found in Note 19, 



(48) 



■p'^^l^ + 'V+Mc- 


■w)]), 


iTCcr 
Now, if we put 




^'_l(w.a') = (9'), 




we find 

^^ -' i7tr[ ' c )' 




and, in virtue of (106), 





(49) 



d'=-|a'-grad(g,'). (50) 



c 



Comparing the formulae (49), (48), (50) and (107) with (79), 
(33) and (34), keeping in mind that, when V is very small, the 

factor 1 — '" may be omitted in the second of the equations (79), 

and replaced by 1 H — - in the numerator of the first, we see that 

there is perfect equality of form. Hence, if we speak of correspond- 
ing states when the dependency of d', h' on x', y , z, t' in a moving 
system is the same as that of d, h on x, y, s, t m a, stationary one, 
we may draw the following conclusion. The field produced at distant 
points of a moving system by an electron whose coordinates x, y, s 
are certain functions of i (the local time belonging to the instantaneous 
position of the electron) corresponds to the field produced in a system 



27 NOTES. 265 

without translation by an equal electron whose coordinates x, y, z 
are the same functions of t. 

Of course, this theorem may be extended to any number of 
electrons, so that we may also apply it to a polarized particle. We 
shall suppose this latter to be so small that the differences between 
the local times of its various parts may be neglected. Then it makes 
no difference, whether we say that the coordinates x , y', z' of an 
electron moving in the particle are certain functions of the local 
time f belonging to the ipstantaneous position of the electron itself, 
or that they are the same functions of the local time belonging to 
some fixed point, say the centre, of the particle, and we have the 
proposition: The field produced in a moving system by an electric 
moment whose components are certain functions of t' (the local time 
of th6 centre of the particle) corresponds to the field existing in a 
system without translation in which there is an electric moment 
whose components are the same functions of t. But, in the latter 
case, the field is determined by (88) and (89). Therefore, we shall 
have for the moving system 

^^ ' iTt \ ox r oy r os r ]' 

[P] 



iTccr 



> 



and we shall find d' and h' by using the formulae (50) and (107). 
It follows from this that the expressions for the field belonging 
to the electric moment represented by (108) may be found as stated 
in the text. 

27 (Page 60). In a stationary system the condition at the sur- 
face of a perfectly conducting body is, that the electric force be at 
right angles to it. This follows from the continuity of the tangential 
components of the force, combined with the rule that in a perfect 
conductor the electric force must be zero, because otherwise there 
would be a current of infinite strength. 

Now, in a moving system, an electron that is at rest relatively 
to it is acted on by a force which, according to (23), is given by 

d + |[w.h]. 

As this is equal to the vector d' defined by (98), d' plays 
exactly the same part as d in a system without translation, and by 
going somewhat further into the phenomena in ponderable bodies, 
one can show that, in a moving system, d' must be normal to the 
surface of a perfect conductor. Moreover, for the fi-ee ether, the 



266 NOTES. 28, 29 

equations which determine d' and h', when referred to moving axes 
and local time, are identical in form with those which we have for 
d and h, when we use axes having a fixed position in the ether. 
This appears at once from the equations (100) — (103). 

28 (Page 62). Since l\, = d^, and h.(^) = — dyj^,, we have 
and for the energy per unit of volume 

^e + ^m=i{ (d, + ^ir))' + (K + Kir))' ) 
= W;+K') +m^r) + Kir))- 

29 (Page 67). Problems relating to the motion of the innumerable 
electrons in a piece of metal are best treated by the statistical method 
which Maxwell introduced into the kinetic theory of gases, and 
which may be presented in a simple geometrical form so long as we 
are concerned only with the motion of translation of the particles. 
Indeed, it is clear that, if we construct a diagram in which the 
velocity of each electron is represented in direction and magnitude 
by a vector OP drawn from a fixed point 0, the distribution of the 
ends P of these vectors, the velocity points as we shall say, will 
give us an image of the state of motion of the electrons. 

If the positions of the velocity points are referred to axes of 
coordinates parallel to those that have been chosen in the metal 
itself, the coordinates of a velocity point are equal to the components 
I, 7], g of the velocity of the corresponding electron. 

Let dA be an element of volume in the diagram, situated at the 
point (I, 7], g), so small that we may neglect the changes of |, rj, g 
from one of its points to another, and yet so large that it contains 
a great number of velocity points. Then, this number may be 
reckoned to be proportional to dX. Representing it by 

filv,QdX (51) 

per unit volume of the metal, we may say that, from a statistical 
point of view, the function f determines the motion of the swarm 
of electrons. 

It is clear that the integral 

ff(^,n,QdX, 

extended over the whole space of the diagram, gives the total num- 
ber of electrons per unit of volume. In like manner the integral 

j\fX^,ri,i)dl (52) 



29 NOTES. 267 

represents the stream of electrons through a plane perpendicular to 
OX, i. e. the excess of the number passing through the plane to- 
wards the positive side over the number of those which go in the 
opposite direction, both numbers being referred to unit of area and 
unit of time. This is seen by first considering a group of electrons 
having their velocity points in an element dX; these may be re- 
garded as moving with equal velocities, and those of them which 
pass through an element da of the said direction between the mo- 
ments t and t + dt, have been situated at the beginning of this 
interval in a certain cylinder having da for its base, and the height 
\^\dt. The number of these particles is found if one multiplies the 
volume of the cylinder by the number (5i). 

Hence, if / means an integration over the part of the diagram 
1 

on the positive side of the tj^- plane, and / an integration over the 

part on the opposite side, the number of the electrons which go to 
one side is 

d6dtfif{^,7],0dX, 

1 

and that of the particles going the other way 

d6dtf-^f{i,r],i)dX. 

2 

The expression (52) is the difference between these values divided 
by dadt. 

If all the electrons have equal charges e, the excess of the 
charge that is carried towards the positive side over that which is 
transported in the opposite direction is given by 

J=efifdl, (53) 

and it is easily seen that, denoting by m the mass of an electron 
and by r^ = l^ + y]^ + ?^ the square of its velocity, we shall have 

\r=^mj\r^fdl (54) 

for the difference between the amounts of energy that are carried 
through the plane in the two directions. The quantities (')3) and (54) 
are therefore the expressions for the flow of electricity and for that 
of heat, both in the direction of OX. 

The function f is deteimined by an equation that is to be re- 
garded as the fundamental formula of the theory, and which we now 
proceed to establish, on the assumption that the electrons are subjected 



268 NOTES. 29 

to a force in the direction of OX, giving them an acceleration X 
equal for all the corpuscles in one of the groups considered. 

Let us fix our attention on the electrons lying, at the time t, 
in an element of volume dS of the metal, and having their velocity 
points in the element dX of the diagram. If there were no encounters, 
neither with other electrons nor with metallic atoms, these electrons 
would be found, at the time t -f dt, in an element dS' equal to dS 
and lying at the point {x -\- Idt, y-\-rjdt, + t<it). At the same 
time their velocity points would have been displaced to an element 
dX' equal to dX and situated at the point (| + Xdt, fj, ^) of the 
diagram, so that we should have 

f(i + Xdt, rj, t, x + ^dt, y+rjdt, s + t,dt, t + dt)dS'dl' 

= /■(!, ri, g, X, y, s, t)dSdl. 

The impacts which take place during the interval of time con- 
sidered require us to modify this equation. The number of electrons 
constituting, at the time t-{-dt, the group specified by dS' and dl', 
is no longer equal to the number of those which, at the time t, be- 
longed to the group {dS, dl), the latter number having to be dimi- 
nished by the number of impacts which the group of electrons under 
consideration undergoes during the time dt, and increased by the 
number of the impacts by which an electron, originally not belonging 
to the group, is made to enter it. Writing adSdXdt and idSdXdt 
for these two numbers, we have, after division by dSdX^dS'dX', 

f{i + Xdt, ri, I, x + ^dt, y + ndt, + tdt, t + dt) 

= f(.h V, t, X, y, 2, t) + {b-a)dt, 

or, since the function on the left-hand side may be replaced by 

m, V, i, X, y, ^, t) + (^x + IL^ + ^, + %t + ^)dt, 

This is the general equation of which we have spoken 
We have now to calculate the values ot a and i. We shall 
simplify this problem by neglecting the mutual encounters of the 
electrons, considering only their impacts against the metallic atoms. 
We shall further treat both the atoms and the electrons as perfectly 
elastic rigid spheres, and we shall ascribe to the atoms masses so 
great that they may be regarded as unmovable. 

Among all the encounters we shall provisionally consider only 
those in which the line joining the centra of the atom and the 
electron has, at the instant of impact, a direction lying within a de- 



29 NOTES. 269 

finite cone of infinitely small solid angle dm. If B, is the sum of 
the radii of an atom and an electron, and ii the number of atoms 
per unit of volume, the number of electrons of the group (51) which 
undergo an impact of the kind just specified during the time dt, is 
equal to 

«i?Y(|, rj, ^)r cos »dl dm dt. {56) 

Here ^ is the sharp angle between the line of centra and the direc- 
tion of the velocity r. 

The velocity of the electron at the end of a collision is found 
by a simple rule. After having decomposed the original velocity into 
a component along the line of centra and another at right angles to 
it, we have only to reverse the direction of the first component. 
Hence, the new velocity point P', whose coordinates I shall call |', 
rf, i,', and the original one (!,»?, t) lie symmetrically on both sides 
of the plane IF passing through at right angles to the axis of 
the cone dco, and when the point P takes different positions in the 
element dX, the new point P' will continually lie in an element dk' 
that is the image of dl with respect to the plane W, and is there- 
fore equal to dl. 

This last remark enables us to calculate the number 6, so far 
as it is due to collisions taking place under the specified conditions. 
By these, a velocity point is made to jump from dX' to dX, and the 
number of these „uiverse" encounters is found by a proper change 
of the expression (5(!). While we replace |, t], g by |', rj', g', we 
must leave the factor r cos d-d I unaltered, for we have dl' = dX, 
r' = r (if r is the velocity whose components are |', t]', &,'), and the 
line joining the centra makes equal angles with r and /. 

We get therefore 

nP?f{i,', r(, ^)r cos&dldadt. 

Subtracting (ofi) from this and integrating the result over all 
directions of the axis of the cone da which are inclined at sharp 
angles to the direction of r, we shall obtain the value of (h — a)dldt. 

When the force which produces the acceleration X has a con- 
stant intensity, depending only on the coordinate x, there can exist 
a stationary state, in which the function f contains neither y nor z. 
For cases of this kind, which occur for instance when the ends of a 
cylindrical bar are kept at different temperatures, or when it is sub- 
jected to a longitudinal electric force, the fundamental equation (55) 
becomes 

nB?rf\a^, V, - m, V, i) } COS * rf« = x|{ + |-g • (57) 

In performing the integration we must leave |, tj, £ unchanged. 



270 



NOTES. 29 



so that r is a constant, but we must not forget that the values of 
I', -ri, g' depend on the direction of the line joining the centra. 
Denoting by f, g, h the angles between this line (taken in such a 
direction that the angle with r is sharp) and the axes, we have 

|'= i, — 2r cosa' cos/; v'—V - ^r cos«' cosg, g'= g— 2r cos'S' cosh. 

So long as the state of things is the same at all points of the 
metal, the electrons will move equally in all directions. It is natural 
to assume for this case Maxwell's well known law expressed by 

/■(I, ri, t) = As-''-% (58) 

where A and h are constants. 
Using the formulae 



— 00 



71 



we find from (58) for the number of electrons per unit of volume 

+ 00 +00 +00 



-00 —00 —CO 



and for the sum of the values of |^, for which we may write JVg^ 
if we use a horizontal bar to denote mean values 



+ 00 +00 +00 



iV|2 = ^JJJa-"(l^ + f + '?)|2^| dri dl = AJ,J,' = T^l/f^ 



-00 — 00 — 00 



It follows from these results that 



Z5 J_ 



l' = t = i' = iT:,: 



and that the mean value of the kinetic energy of an electron is 

equal to 

3m 

But we have already made the assumption that the mean kinetic 
energy is equal to aT. Therefore 



29 NOTES. 271 

an equation which, conjointly with (5!>), tells us in what manner the 
constants h and A are determined by the temperature and the number 
iV" of corpuscles per unit of Tolume. 

It is clear that the formula (58) can no longer hold when there 
is an external force or when the ends of a metallic bar are unequally 
heated. Yet, whatever be the new state of motion, we shall always 
have a definite number N of electrons per unit of volume, and a 
definite value of the mean square of their velocities, and, after having 

3 
assigned to h and A such values that , is equal to this mean square 

and J.l/y3 to the number N, we may always write 

m,ri,^) = Aa-'"- + cp(i,,l,t), (01) 

where qo is a function that remains to be determined. For this we 
have the fundamental equation (y/'J and in addition to it the con- 
ditions 

fq>dX = 0, f(pr''dl = Q, (62) 

which must be fulfilled because the term A£~'"''' has been so chosen 
that it leads to the values of N and r^ really existing. 

The function cp is the mathematical expression for the change 
which an external force or a difference of temperature produces in 
the state of motion of the system of electrons. Now, this change 
may be shown to be extremely small in all real cases, so that the 
value of (p is always small in comparison with that of A£~'"^. 
Hence, on the right-hand side of equation (57) we may replace f 
by Ae~'"^. On the left-hand side, on the contrary, we must use the 
complete function (61), because here we should find zero, if we 
omitted the part cp(^, rj, ^). 

The equation therefore becomes 

nB'rf{<p(i', 7]', t) - -jPd, V, 0} cos & dm 

= (-2hAX + ^-r^A'^)U-'-\ (63) 

Let us try the solution 

<p(^,v,t) = h(r), m 

where ;t is a function of r alone. This assumption is in accordance 
with the conditions (62), so that we have only to consider the prin- 
cipal equation (63). Substituting in it the value (64) we first find 

f{cp(^', r(, r) - ^(%, V, t)\ 008 & dm = x(r)f(^-l) cos 9 dm 
= _ 2rx(r) f COST'S' cos f da. 



272 NOTES. 29 

Let us imagine two lines OP and OQ, drawn from the origin 
of coordinates, the first in the direction of the velocity (|, ij, ^), and 
the second in that of the line of centra at the moment of impact, 
the angle POQ = & being sharp. Denoting by /a the angle POX 
and by ip that between the planes POX and POQ, we have 

cos f = cos ju, cos -9- + sin ft sin %■ cos i), 

1" in 

I cos^d-cosfda = I I cos^&(cosii(iosd--\-smiism&cosil>)siiid-d&df 





1 



= 2n cos [I I COST'S' siR&d& = y^ cos ft = -— ^~7 



by which (63) assumes the form 

- ^nBnrxir) = (- 2hAX + ^- r'A^)^s-"'^, 

showing (because | disappears on division) that our assumption reaUy 
leads to a solution of the problem. 
If we put 

^J 7 

the result is 

x(r)^l{2hAX-^ + r^A^)^s-\ {65) 

Finally we find from (53) and (54) for the currents of electricity 
and of heat 

J=efi'x(r)dl, 

W=^mJ\r^l{r)d}.. 

In these formulae |^ may be replaced by \r^ and dk by Anr'^dr] 
the integration is thereby reduced to one with respect to r from 
to oo. Next substituting the value (^5), and choosing s = r" as a 
new variable, we are led to the integrals 

00 00 00 

fss-'"ds, Jsh~'"ds and fsh-'"ds. 



The values of these are 

J 2 6 



29 NOTES. 273 

so that the two currents are giTen by 

The coefficient of electric conductivity is easily found from 
the first of these equations. Let the cylindrical bar be kept at uni- 
form temperature throughout its length. Then -j— = 0, -5— = 0, and 
when there is an electric force E producing an acceleration 

eE 



X 

VI 



the electric current will be 



3hm 
We conclude from this that 

^~ Shm ' 

or, if we use the relations (59) and (60), introducing at the same 
time a Telocity u whose square is equal to the mean square r\ so 
that m == — 5- , 

2" eHNu 



Sx ccT 

In order to find the conductivity for heat we shaU consider a 
bar between whose ends a difference of temperature is maintained, 
these ends being electrically insulated, so that no electricity can enter 
or leave the metal. Under these circumstances the unequal heating 
wiR produce a difference of potential which increases until the electric 
force caRed forth by it makes J vanish. The final state will be 
characterized by 

giving 

„^ 2 , 4 dh StcIAcc dT 

where we have also used the relation (60). From this we infer the 
coefficient of thermal conductivity 



, SnlAa 8 t/ 2 ,„ 



It remains to add that the quantity I may be considered as a 
certain mean length of free path. 

Lorcntz, Theory of electrons. 



18 



274 NOTES. 30 

30 (Page 77). As a preliminary to the deduction of Wien's 
law, we shall extend to the case of an oblique incidence the reasoning 
given in § 46. A beam of light propagated in a direction lying in 
the plane XOZ and making an angle ■& with OX may be represented 
by expressions of the form 



/, a; cos ■9- + « sin •9- , \ 
a COS nu ■ h pj , 



and when it faUs upon a fixed mirror whose surface coincides with 
the plane YOZ, we shall have functions containing the factor 



/, , a;cosa)' — asin* , \ 
cosnU-\ h 2^j 



for the quantities relating to the reflected light. 

Now, the theorem of corresponding states (§ 45, Note 26) teUs 
us that when the mirror has a translation with velocity w in the 
direction of OX, there can be a state of things represented by equa- 
tions in which the above goniometric functions are replaced by 



x' COS'9' -f- « sin'* 
and 



/,, X cos'9' 4- a sm* , \ /^^n 

coswU — ■ hP) {o6) 



I ., , a;' cos ■9' — a sin ■9' , \ ^„„-, 

COS n(t -i -^ hi'j, (67) 

where 

X = X — wt and t = t vX . 

The frequencies of the beams are given by the coefficients of t 
in these expressions {6€) and (67) 

wll H cos'9') and w(l cos^l, 

so that, if the frequency of the incident rays is 

n\\. -(- - cos'9') = n, 

that of the rays reflected by the moving mirror is given by 

nil coS'S'j. 

It follows from this that a wave-length I is changed to 

a(i+^"coS'&). 

We shaU also have to speak of the pressure acting on a per- 
fectly reflecting mirror receiving under the angle ■^ a bundle of 



30 NOTES. 275 

parallel rays. As it will suffice to know the pressure exerted on 
the mirror when at rest, we may apply the formula found in § 25. 
Since aU. the light is reflected, we have s"=0, and |s'| = |s|, the 
magnitude of these last vectors being equal to the product by c of 
the energy i existing in the incident beam per unit of volume. 
Moreover, if A is the area of the mirror, we have Z:= If = A cos d: 
As the vectors s and s' are in the direction of the rays, it is easily 
seen that the vector s — s' is directed towards the mirror alons the 
normal. The resultant force is therefore a normal pressure whose 
magnitude is 2Aicos^d', or 2*cos^'9' per unit of area. 

Turning now to the proof of Wien's law, we shall consider a 
cylindrical vessel closed by a movable piston and void of ponderable 
matter. We shall conceive the internal space to be traversed in aU 
directions by rays of light or heat, it being our object to examine 
the changes in iatensity and wave-length that are brought about by 
the motion of the piston. We suppose the latter to be perfectly 
reflecting on the inside, whereas the walls and the bottom of the 
cylinder are „perfectly white"^ by which we mean that they reflect 
the rays equally in aU directions and without any change in wave- 
length or any loss of intensity. By making these assumptions, and 
by supposing the motion of the piston to be extremely slow, we 
secure for aU instants the isotropy of the state of radiation. 

Let us fix our attention on the rays existing at a certain time t 
with wave-lengths between the limits 2. and J. + dX, and let us de- 
note by ip (^X) dX the energy per unit of volume belonging to these 
rays, or, as we shall say, to the group (A, X -\- dX). If A is the 
surface of the piston and h its height above the bottom of the 
cylinder, the total energy belonging to the group in question is 

J=Ahil){X)dX, (68) 

and we may find a differential equation proper for the determination 
of ^ as a function of X and t, by examining the quantities of energy 
that are lost and gained by the group (A, X -\- dX). 

In the first place a loss is caused by the reflexion of part of 
the rays against the moving piston, for every ray which faUs upon 
it, has its wave-length changed, so that, after the reflexion, it no 
longer belongs to the group (X, X + dX). In order to calculate the 
loss we may observe that the rays of which we are speaking are 
travelling equally in aU directions; hence, if we confine ourselves to 
those whose direction lies within an infinitely narrow cone of solid 
angle da, we have for the energy per unit of volume 

— j/>(A)^co dX, 

18* 



276 NOTES. SO 

and for those rays whose direction makes an angle between 9- and 
& -{- d& with the normal to the piston (drawn towards the outside) 
the corresponding value is 

^^(;i) 8in& dd'dX. 

During the interval dt the piston is struck by these rays in so 
far as, at the time t, they were within a distance c cos 9- dt from the 
piston, i. e. in a part of the cylinder whose volume is c A cos 9- dt, 
so that the energy falling upon the piston is 

'l-cA(p(l) sins' coa9d9dXdt. 

Integrating from 9 = to 9=1-71, one finds for the energy 
that is lost by the group (A, A + dl) 

\cA^{l)dkdt. (69) 

On the other hand a certain amount of energy is restored to 
the group because rays originally having another wave-length, get 
one between 2 and 1 -f dX by their reflexion against the moving 
piston. 

Let us begin by especially considering the rays whose direction 
before reflexion is comprised within a cone dm whose axis makes 
an angle ■9-(< -^n) with the normal to the piston. If k' is their 
wave-length before reflexion, it will be changed to 

X = (l + ^^ cos 9^ k', 

where w is to be reckoned positive when the motion of the piston 
is outward. Hence, if the new wave-length is to lie between k and 
X + dX, the original one must be between X' and X' -f dX', where 

r=(l-^|'coss)l, 
dX' = (l — - cos 9] dX. 
The energy of these rays per unit of volume is 
i = —tl!{X')d(odX' 

and one sees by a reasoning similar to that used above that the 
amount of energy belonging to the group of rays defined by da, X', 
dX', which falls upon the piston during the time dt, is equal to 

cAi cos 9 dt. 

Part of this energy is spent in doing work on the piston, and 
it is only the remaining part that is gained by the group (X, X-{-dX). 



30 NOTES. 277 

The pressure exerted on the piston by the rays of which we 
are now speaking being 

and its work during the time dt 

2wAi aos^^dt, 

the amount of energy restored to the group {X, l + dX) is given by 

cAi^08^dt—2ioAicQ'&^%dt. (70) 

As we constantly neglect the square of w, we shall replace I 
in the second term by 

^ij{X)d(odk 

and in the first term by 

J- (1 — - coS'&j t{X') da dl 



,^(r) = i^(/i)- — cos^./i-^''' 



smce 

By this we get for the expression {70) 

^\QOB»--,ij{k)-^-^coB^&[2i>{l) + ).^y\dmdXdt. 

Extending the integral of this with respect to d(o over aU direc- 
tions of the rays for which 9- <i\-n, we find the energy that is 
restored to the group {I, X -f- dX) and must be subtracted from {69). 
Since 

I cos d'dco = :t, I cos^&dc3 == ^^, 

the result of the integration is 

and we have for the change of the energy existing in the cylinder, 
so far as it belongs to wave-lengths between X and X -\- dX, 



dJ 



= - \ wA\2tl>{X) + l^^^\dX. 



But, since -tt = w, we see from (68) that 
^ = wAxlj{X)dX + Ah^dX, 



278 NOTES. 30 

so that 



, , J dil> 2 , 1 .dip 



or, if we put 



w , 
1« 



-/<5^ + A-^,t). (71) 



This differential equation enables us to calculate the change 
which the motion of the piston produces in the distribution of the 
energy over the different wave-lengths. In order to put it in a form 
more clearly showing its meaning, we shall first deduce from it the 
rate of change of the total energy per unit of volume 



K 



= fi} dl. 



For this purpose we have only to multiply (7t) by dl, and to 
integrate each term from A = to A = oo. Since 



CO 



dip -. _dK 



and 



we find 



00 OO 00 

jl^dl= lA^I^^J- jtljdX = - I ipdX = ~K, 



In deducing this equation I have supposed that for A = oo the pro- 
duct X^) tends towards the limit 0. 

Now, when the velocity w is given for every instant, k is a 
known function of the time and so will be K. We may therefore 
introduce this latter quantity as independent variable instead of t. 
Putting 

and considering V as a function of this quantity and of 

log A = 92, 
we find from (71) after division by —hip, 



31 NOTES. 279 

This is simplified still further if, instead of | and t], we introduce 

I' = I and ri =1 + Ari 
as independent variables. The equation then becomes 

gl^Ciog^.- -5-0 = 0, 

showing that the expression 

log?/;- i'-|'=log(i/'^^'), 
and therefore 

itself must be a function of >/ alone. But 

,2'=| + 47; = 41og(A^ij, 

so that fK't may also be represented as a function of XK^^. The 
solution of our equation is thus seen to be 

K-irp{k,K) = F(},Ki), (72) 

where we have expressed that ^ is a function of A and K, and where 
the function F remains indeterminate. 

If, in the course of the motion of the piston, the value K' of K 
is reached, we shall have, similarly to (73), for any wave-length 

The right-hand side of the first equation becomes equal to that of 
the second, if we replace A by 

so that 

Hence, if in the original state the distribution of energy is given 
by the function <p(l), i. e. if, for aU values of 2, 

we find for the corresponding function in the final state 

31 (Page 80). Planck finds in C. G. S.- units 
« = 2,02-10-i«, 
(so that the mean kinetic energy of a molecule would be 2,02 • 10" ^^ T 
ergs), for the mass of an atom of hydrogen 

1,6 ■ 10~^* gramm 



280 NOTES. 32, 33 

and for tlie universal unit of electricity expressed in the units whicli 
we have used 

1,6- lO-socyi^ 
(see § 35). 

32 (Page 81). In a first series of experiments Hagen and 
Rubens deduced the absorption by a metal from its reflective power; 
they found that for A = 12/i, S(i and even for X = 4fi the results 
closely agreed with the values that can be calculated from the con- 
ductivity. In later experiments made with rays of wavelength 25,5 j^ 
(„Reststrahlen" of fluorite), which led to the same result, the emis- 
sivity of a metal was compared with that of a black body, and the 
coefficient of absorption calculated hy our formula (122) (p. 69). 

33 (Page 81). Let us choose the axis of x at right angles to 
the plate, so that ic = at the front surface and a; =z/ at the back; 
further, let a be the amplitude of the electric vibrations in the in- 
cident beam, this beam being represented by 



6^ = a cosn(t — ^ + pj ■ 



The electric force E in the interior of the thin plate may be 
considered as having the same intensity at all points. It produces a 
current of conduction 

K - ^^. 

and a dielectric displacement in the ether contained in the metal. The 
variations of this displacement, however, do not give rise to any ther- 
mal effect, and the heat produced will therefore correspond to the 
work done by the force E while it produces the current i . Per 
unit of time and unit of volume this work is equal to 

so that the development of heat in a part of the plate corresponding 
to unit of area of its surface is given by 

Now, at the front surface, E^^ is equal to the corresponding 
quantity in the ether outside the metal (on account of the continuity 
of the tangential electric force), i. e. to d^, -f d^^^, where dj,(^j relates 
to the reflected beam. Since, however, the amplitude of d, ,^. is propor- 
tional to z/, and since we shall neglect terms containing zl^, we may 
omit dy^^j. In this way we find for the development of heat 

(ja"z/ coa^{nt-\-p), 



34, 35 NOTES. 281 

and for its mean value during a time comprising many periods 

The coefficient of absorption A is found if we divide this by the 
amount of energy -^-a^c which, per unit of time, falls upon the portion 
of the plate considered. 

34 (Page 85 j. This is confirmed by the final formula for a/ 
(p. 89), according to which this quantity is proportional to s^, and 

therefore to -p- 

35 (Page 87). The truth of this is easily seen if we consider 
both the metallic atoms and the electrons as perfectly elastic spheres, 
supposing the former to be immovable. Let a sphere whose radius I{ 
is equal to the sum of the radii of an atom and of an electron be 
described around the centre of an atom, and let a line OP be 
drawn in a direction opposite to that in which an electron strikes 
against the atom. Then, the position of the point Q on the sphere 
where the centre of the electron lies at the instant of impact may 
be determined by the angle POQ ==^ & and the angle qp between the 
plane POQ and a fixed plane passing through OP. The probability 
that in a collision these angles lie between the limits -S' and & + dd', 
(p and (p + d(p, is found to be 

~ sin ■9- cos %-d%'d(p, (73) 

where %■ ranges from to \%, and cp from to 2n. 

Let us also represent the direction in which the electron re- 
bounds, by the point S where a radius parallel to it intersects the 
spherical surface. The polar coordinates of this point are &' =^2%- 
and ip' = <p, and if these angles vary between the limits &■' and 
Q-' _j_ cl&\ qp' and q)' + d^', the point S takes aU positions on the 

element 

da = It' sin d^'d 9' d<p' 

of the sphere. But we may write for the expression (7.3) 

-— Bind-' d&' dm', 

so that the probability of the point S lying on the element dd is 

da 

This being independent of the position of d6 on the sphere, we 
conclude that, after an impact, all directions of the velocity of the 
electron are equally probable. 



282 NOTES, 36 

36 (Page 88). Considering a single electron whicli, at the time t, 
occupies the position F, we can fix our attention on the distance 
PQ = l over which it travels before it strikes against an atom. If 
an electron undergoes a great number iV of collisions in a certain 
interval of time, we may say that the experiment of throwing it 
among the atoms and finding the length of this free path I is made 
with it N times. But, since the arrangement of the atoms is highly 
irregular, we may just as weU make the experiment with N different 
electrons moving in the same direction with a common velocity u. 
Let us therefore consider such a group, and let us seek the number N' 
of it, which, after having travelled over a distance I, have not yet 
struck against an atom, a number that is evidently some function of I. 
During an interval dt a certain part of this number N' wiU be dis- 
turbed in their rectilinear course, and since this part wiU be pro- 
portional both to N' and to dt, or, what amounts to the same thing, 
to the distance dl = udt, we may write for it 

^N'dl, (7i) 

where /3 is a constant. Hence, while the distance dl is travelled 
over, the number N' changes by 

dN' = - ^N'dl, 
so that we have 

N' = iVf-/*', 

because N' = N ior I = 0. 

The expression (74), which now becomes 

^N£-?'dl {75) 

gives the number of electrons for which the length of path freely 
travelled over lies between I and I -|- dl. The sum of their free 
paths is 

^NlE-fhll, 

and we shall find the sum of all the free paths if we integrate from 
1 = to 1= oo. Dividing by N, we get for the mean free path 

00 

L = &ll£-^'dl-=^- 



The number (^75) of free paths whose lengths lie between I and 
I -\- dl is therefore equal to 



1 



, Ne 'm dl, 

I'm 



87—39 NOTES. 283 

or, since 



-»T 'u& 



equal to 

u9 



£ 



'm dl. 



37 (Page 89). This case occurs when the atoms and the elec- 
trons are rigid elastic spheres, the atoms being immovable, for it is 
clear that an electron may then move with different velocities in 
exactly the same zigzag line. Other assumptions would lead to a 
value of Z^ depending on the velocity u, but then we should also 
have to modify the formula given in § 50 for the electric conductivity. 

The final formula for . would probably remain unaltered. 

38 (Page 90). It may be noticed that the numbers given in 
Note 31 can be said to be based on the formula (148), if in calcu- 
lating them one uses only the part of the radiation curve correspond- 
ing to long waves. 

39 (Page 92). According to what has been said, the potential 
and the kinetic energy may be represented by expressions of the 
form 

immediately showing that the amounts of energy belonging to each 
of the n fundamental modes of vibration have simply to be added. 
Since for small vibrations the coefficients a and i may be regarded 
as constants, each mode of motion is determined by an equation of 
Lagrange 

dtXdpJ dp I,' 

or 

^kPk = - "-kPk, 

the general solution of which is 

p^= a cos iy '^j + ^y 

where a and /? are constants. 

In this state of motion there is a potential energy 



'^a,p,^=\aycos^{]/lt + ^), 



284 NOTES. 40 

and a kinetic energy 

^&,i)/=^-a,«^sin^(]/|;<+^), 
both of which have the mean value 

40 (Page 94). Taking three edges of the parallelepiped as axes 
of coordinates, and denoting by f, g, h the direction cosines of the 
electric vibrations of the beam travelling in the direction {fi^, fi^, ftg), 
we may represent this beam by the formulae 

d, = fa cos n(t- f'^^+J^y+J^ + p), 

d^ = ga cos n (t - ^^^ + M+ ^ + p)^ 

d, = ha cos n (t - f^i^_f^i^ +^). 

If we assume similar formulae with the same constants a and p for 
the seven other beams, replacing fi^, fig, ftj by the values indicated 
in (149), and f, g, h by 

f,-9,- M -f,9,~ h- -f,-g, /j; 
f,-9,— M -f,'J,- '*; ~f,-9,h; 
f, 9, ^» 
respectively, the total values of d^, d , d^ are given by 



^ = — »/a cos -^^^~- sm - - - sm — --- cos n{t -\- p), 



c c c 



J, = — o^a sm --— cos -^-^ sm -^- cos n{t -\- p), 

do, • nii.x ■ nii.y na,z ,, , , 

J = — 8Kasm--;-sm~^-^cos-^ -- cosw(^ + p)- 



(76) 



By these the condition that d be normal to the walls is fulfilled 
at the planes XOY, YOZ, ZOX, for at the first plane, for example, 
2 = 0, and consequently d^ = 0, d, =0. 

The same condition must also be satisfied at the opposite faces 
of the parallelepiped. This requires that, if g^, q^, q^ have the meaning 
given in the text, 

sin^'''^ = 0, sin^^ = 0, sin^^ = 0. 
c ' c ' e 

Therefore, 

^_9i :'iif^fe ii'f^sla 
c c c 



41 NOTES. 285 

must be multiples of jt, and since — = — > 



must be whole numbers. 



I ' X ' I 



41 (Page 94). If one of these states, say a state A, is deter- 
mined by the formulae {76) of the preceding Note, in which f, g, h 
relate to any direction at right angles to the direction (n^, fi^, fig), 
a state of things A' in which the polarization is perpendicular to 
the former one is represented by equations of the same form (with 
other constants a and jo'), in which f,g,h are replaced by the con- 
stants f, g , h' determining a direction at right angles both to {f, g, 
h) and to (/ii, /i^j fa)- It is easily seen that any other mode of 
motion represented by formulae like (76) with values of f, g, h such 
that 

may be decomposed into two states of the kind of A and A'. The 
total electric field will therefore consist of a large number of fields 
A and A', each having a definite amplitude a and phase p. In 
order to find the total electric energy we must calculate for each 
mode of motion the integral 



I,/' 



and for each combiuation of two modes 

J[6-A')dS (77) 

Now, it may be shown that all the integrals of the latter kind 
are zero. For a combination of two states such as we have just now 
called A and A' ( which are characterized by equal values of fij , (tg ? 
fig and of the frequency n), this is seen if one takes into account 
that in the integrals 

'h 'A 1-. 

I cos^^^^^ dx, JBija^^^^dx, I cos^^-^ dy eic. (78) 



the square of the cosine or the sine may be replaced by 4, so that 
(77) becomes 

8(/"r + 99 + hJi') aa'q^q^q^ cos n(t ■+ p) cos n (t + p), 

which is because the directions (f, g, h) and (f\ g , li) are at right 
angles to each other. 



286 NOTES. 42 

In any other case at least one of the coefficients — -i, — ^, —^ 

will be different for the states d and d'. Thus, -'^' may have the 
value h for one state and the value Jc' for the other. The integrals 

/ COS kx cmh'xdx = :j ^ . ^,, sin (Ic + h') q^ + ^(k — k') ^"^(^'^ ~ ^'^'^^i' 



/ sinAia; sink'xdx = - g^^., y- sin(A; + lc')q^ + ^^j^_j.'^ sin (A; — /(;')9'i 



both are zero, because Jcq^ and fe'^j are multiples of jr. Consequently, 
each of the three integrals 

fij'JS, etc. 

iato which (77) may be decomposed vanishes. 

It is readily seen that similar results hold for the magnetic 
energy. It will suffice to observe that, in the state represented by 
the formulae (76), the magnetic force has the components 



iu.,!ic nii,y nii„z 

— — cos - - cos — ^=^5- I 

c c c 



^.= 8f a sm -^— COS — - cos — ^=^=5- sin n(t -\- p), 



,, = Sg a COS — - - sm -^-~- cos -^'~ sm w (f + 2V> 
.= 011 acos— ~— cos-^^-^ sin -^^^— sinn(t + p), 

C C C \ 1 i /; 

where 

/■'= jttg/i - [i^g, ()'= fi^f— fi^h, h' = [i^g - jxg/' 

are the constants determining a direction perpendicular both to (ftj, 
,it2, fij) and to (f,g,h). 

If, further, one takes into account what has been said of the inte- 
grals (78), it will be found that the parallelepiped contains an amount 

4(r+ /+ h^) q^q^q^a^ cos^ n(t + p) = 4: q^q^q^a^ aos^ n (t + p) 

of electric, and an amount 

Aq^q^q^a^s\o?n(t-{-p) 

of magnetic energy. Each of these expressions has the mean value 

2q^q^q^a\ 

42 (Page 97). On further consideration I think that it will be 
very difficult to arrive at a formula different from that of Jeans, 



42 NOTES. 287 

so long as we adhere to the general principles of the theory of 
electrons as set forth in our first chapter. But, on the other hand, 
it must be observed that Jeans's theory is certainly in contradiction 
with known facts. Let us compare, for example, the emissivity E^ 
for yellow light of a polished silver plate at 15" C. with that (eJ) 
of a black body at 1200" C, confining ourselves to the direction 
normal to the plate. Silver reflects about 90 percent of the in- 
cident light, so that the coefficient of absorption of the plate is ^, 
and by Kirchhoff's law, E^ = f^E^, if E^ denotes the emissivity of 
a black body at 15". But, by Jeans's theory (see § 74) the emissivity 
of a black body for light of a given wave-length must be proportional 

to the absolute temperature, so that we have E^ =--~E2 = -rE^, 
and£,= f^E,. 

Now, at the temperature of 1200", a black body would glow 
very brilliantly, and if the silver plate at 15" had an emissivity only 
fifty times smaller, it ought certainly to be visible in the dark. 

It must be noticed that we have based our reasoning on Kirch- 
hoff's law, the validity of which is not doubted by Jeans. In fact, 
the point in the above argument was that, at temperatures at which 
a black body has a perceptible emissivity for the kind of rays con- 
sidered, it can never be that, for some other body, only one of the 
coefficients E and A is very small. The silver plate might be ex- 
pected to emit an appreciable amount of light, because its coefficient 
of absorption shows that in reality the exchange of energy between 
its particles and the ether is not extremely show. 

From facts like that which I have mentioned it appears that, 
if we except the case of very long waves, bodies emit considerably 
less light, in proportion to their coefficient of absorption, than would 
be required by Jeans's formulae. The only equation by which the 
observed phenomena are satisfactorily accounted for is that of Planck, 
and it seems necessary to imagine that, for short waves, the connec- 
ting link between matter and ether is formed, not by free electrons, 
but by a different kind of particles, like Planck's resonators, to 
which, for some reason, the theorem of equipartition does not apply. 
Probably these particles must be such that their vibrations and the 
effects produced by them cannot be appropriately described by means 
of the ordinary equations of the theory of electrons; some new as- 
sumption, like Planck's hypothesis of finite elements of energy will 
have to be made. 

It must not be thought, however, that all difficulties can be 
cleared in this way. Though in many, or in most cases, Planck's 
resonators may play a prominent part, yet, the phenomena of con- 
duction make it highly probable that the metals at least also contain 



288 NOTES. 43—45 

free electrons whose motion and radiation may be accurately described 
by our formulae. It seems difficult to see why a formula like Planck's 
should hold for the emission and absorption caused by these particles. 
Therefore, this formula seems to require that the free electrons, 
though certainly existing in the metal, be nearly inactive. Nor is this 
all. If we are right in ascribing the emission and the absorption by a 
metal to two different agencies, to that of free electrons in the case 
of long waves (on the grounds set forth in § 60), and to that of 
„resonators" in the case of shorter ones, we must infer that for inter- 
mediate wave-lengths both kinds of particles have their part in the 
phenomena. The question then arises in what way the equilibrium 
is brought about under these complicated circumstances. 

It must be added that, even in the case of long waves, there 
are some difficulties. To these attention has been drawn by J. J. 
Thomson.^) 

I shall close this discussion by a remark on the final state that 
is required by Jeans's theory. I dare say that it will be found 
impossible to form an idea of a state of things in which the 
energy would be uniformly distributed over an infinite number of 
degrees of freedom. The final state can therefore scarcely be thought 
of as really existing, but the distribution of energy might be con- 
ceived continually to tend towards uniformity without reaching it in 
a finite time. 

43 (Page 102). In Zeeman's first experiments it was not found 
possible neatly to separate the components; only a broadening of the 
lines was observed, and the conclusions were drawn from the amount 
of this broadening and the state of polarization observed at the 
borders. 

44 (Page 110). For great values of the coordinates, the coeffi- 
cients c might be functions of them. They may, however, be treated 
as constants if we confine ourselves to very small vibrations. 

45 (Page 112). The result of the elimination of gj, gg, . . . g^ 
from the equations (176) is 






= 0. {79) 



1) J. J. Thomson, The corpuscular theory of matter, p. 85. 



46 NOTES. 289 

Developing the determinant we get in the first place the principal 
term 

77 = (/; - m,ir) (/, - >n,n'^) . . . (f^- m,y), 

and in the second place terms containing as factors two of the 
coefficients c. These coefficients being very small, we may neglect 
all further terms which contain more than two factors of the kind. 
One of the said terms is obtained if, in the principal term, the two factors 
fk — »»i w^ and /; — m^n^ are replaced by — in c^., • inc,^ = — n^.^. Hence, 
denoting by 77,,, the product which remains when we omit from 11 
the factors f^— iHf.n^ and fi — niin^, we may write for (7.9) 

n-n^:^cln,,= 0, (80) 

1.1 

an equation that can be satisfied by values of w' differing very little 
from the roots n^^, n^^, . . . n^^ of the equation 

JT=0, 

which are determined by (172). 
Thus there is a root 

n" = V + d, (81) 

where S is very small. Indeed, if this value is substituted in (80), 
we may replace n^ by w^^ in all the products 77,.,, and the same may 
be done in the factors of the first term U, with the exception only 
of /"j — '>ny, for which we must write — m,^S. By this 77 becomes 

-m,Sn,(n,'), 

where the last term means the product 77 after omission of the said 
factor, and substitution of n^ = n^ in the remaining ones. 

lu the sum occurring in (86) only those terms become different 
from zero, in which the factor f,. — m^ (corresponding to the par- 
ticular value we have chosen for li) is missing. Our equation there- 
fore assumes the form 

- m,dn,(n,') - n,':2cl.,n,,(n,') = 0, 

from which the value of d is immediately found. 

This value may be positive or negative, but, as it is very small, 
the right-hand side of (81) is positive in any case, and gives a real 
value for the frequency 

46 (Page 113). Equation (79) is somewhat simplified when 
we divide the horizontal rows of the determinant by ]/wj, Yiti^jetc, 
and then treat the vertical columns in the same manner. Putting 

Lorentz, Theory of electrons. 19 



290 NOTES. 



so that 

and using (172), one finds 



- ^'»e^a; - ^^Cf,2, 



= e. 



— tne. 



tne. 



%-' 



= 0. 



46 

m 

(83) 

m 



Let us now suppose that a certain number h, say the first Tt, 
of the frequencies n^, n^, . . . hare a common value v, and let us seek 
a value of v satisfying the condition (84), and nearly equal to v. 
When n has a value of this kind, all the elements of the determinant 
vpith the exception of nl ^ ^ — w^, . . . n ^^ — n^ are very small quantities. 
Therefore, the part which contains these fi — k elements, namely the part 



K-.1-"") ••■ (V-^') 



greatly predominates. We shall therefore replace (84) by 



Mj^— n^, 


— ine^^, 


. -ine,. 


— ine^i, 


Wa^ - n^, 


- ine,. 


-ine,^, 


-ine,„ 


. V - n'' 



tne. 



^me^.l 



ine,.9 



■me. 



— «me„ 



0. 



Finally, since the quantities e are very small, we shall replace n 
by V wherever it is multiplied by an e, so that we find 



-iveji, v^ 






-ive,. 



IVCiv 



— tve^ 

— ive„ 



v' — n^ 



= 0. 



(85) 



an equation of degree Ic in n^. 

Now, on account of the relations (83), the latter determinant 
is not altered when we change the signs of all the elements con- 



46 NOTES, 291 

taining an e (the effect being merely that the horizontal rows become 
equal to the original vertical columns). Hence, after development, 
the equation can only have terms with an even number of these ele- 
ments, so that it is of the form 

(v^- w2)*+ P,{v'- - n^y-'^+F,{v^- n^y-*+- • ■ =0, (86) 

where P^ is made up of terms containing two factors of the form 
ive, Pg of terms containing four such factors, and so on. 

It follows from this that the coefficients P are real quantities. 
But we may go further and prove that, if v^ — v? is considered as 
the unknown quantity, all the roots of the equation (6'5) or (.s'6) 
are real. 

For this purpose we observe that, on account of (85), if we take 
for 1'- — H^ one of its roots, the equations 

(v^— n^^x^— ivej^^x^— ■ ■ ■ — lve^^x,.= 0, 

— ivc.,.^ a;^ + {v^— n^)x^ _ . . . _ ive^^X/. = 0, 

— i ve^i x^ — ivc,.^x^ —...-[- (v2_ n^^x,^ = 

may be satisfied by certain values of x^, x^ . . . %, which in general 
will be complex quantities. Let x^fX^, ■ ■ ■ x^ be the conjugate values. 
Then, multiplying the equations by x^, x^, . . . x^ respectively and 
adding, we find 

[v^- n^) ^y^jXj- vJ:i{e^iXiXj+ e^jX^x,) = 0. (67) 

Now, putting 

we have 

^i^'j = ^/ + ''A 
and, in virtue of {83), 

i.(ej,x,x^+ e,jXjx;) = 2ejt{^,r]^- ^jTj,). 

The two sums in (87) are therefore real, and v^ — n^ must be so 
likewise. 

We have now to distinguish the cases of /c even and Jc odd. 
In the first case (86) is an equation of degree i /,-, when (v^ — n^Y is 
considered as the quantity to be determined, and, since v^' — n^ must 
be real, its roots are all positive. Calling them a^, ^^, y% . . ., we 
have the solution 

n^— v^= ± a, + /3, +}',■■■ 
whence 

n = v±^, v + 2-^' ^±{v'' ■ (**) 

being l values of the frequency. 

19* 



292 NOTES. 47, 48 

When li is odd, equation {86) has the factor v^ — n^, so that one 
root is 

n = V , 

corresponding to the original spectral line. After having divided the 
equation by v^ — n^, we are led back to the former case, so that 
now, besides n = v, there are li — 1 roots of the form {88). 

In the particular case of three equivalent degrees of freedom, 
equation {86) becomes 

{v^ — ny+ (v^- ^2)0.2(6,3632+ egiCia + e,^e.^^) = 0, 

giving n^ — v^ = and 



n^-v^= + vyel^+el^+e\^, 



from which (177) immediately follows, if we replace v by n^ and fgg, 
^si) ^12 ^y tlieir values {82). 

I am indebted to a remark made by Dr. A. Pannekoek for the 
extension of the foregoing theory to cases of more than three equi- 
valent degrees of freedom. 



^b'- 



47 (Page 113). That the distances between the magnetic com- 
ponents of a spectral line wiU be proportional to the intensity of 
the magnetic field (for a given direction of it) is also seen from 
the general equation {86). It suffices to observe that each quantity 
e is proportional to | H j. Therefore Pj is proportional to H^, F^ to 
H*, and so on. The values of n^ — v^ which satisfy the equation 
vary as j H ' itself, and as they are very small, the same is true of 
n — V. 

48 (Page 120). In the following theory of the vibrations of a 
system of four electrons we shall denote by a the edge of the tetra- 
hedron in the position of equilibrium, by I the distance from the 
centre to one of the edges, by r the radius of the circumscribed 
sphere, and by Q' the angle between the radius drawn towards one 
of the angles and an edge ending at that angle. We have 



cos -9- 



-V\' ^ = i«l/2, r==\ayQ. 



In the state of equilibrium one of the electrons A is acted on 
by the repulsions of the three others, each equal to 






and by the force due to the positive charge. The latter force is 



48 NOTES. 293 

the same as when a charge e = -^nr^Qf, were placed at the point 0. 
Hence we have the condition of equilibrium 



or 



3e' 


cos 


# + 


iitr 




Qo 


= — 


3e 



0, 



The frequency of the first mode of motion is easily found by ob- 
serving that, after a displacement of all the electrons to a distance 
r -{■ 8 from the centre, where 8 is infinitely small, the resultant 
force acting on any one of them would remain zero, if the attraction 
exerted by the positive sphere were still equivalent to that of a 
charge e at 0. As it is, there is a residual force due to the at- 
traction of the positive charge included between the spheres whose 
radii are r and r -\- 8. The amount of this charge being 4:%r'^Q8, 
and the force exerted by it on one of the electrons eQ8, we have 
the equation of motion 

in-iu^=eQ8, 
giving for the frequency 

n^= -■ 

m 

Let us next consider the motion that has been described in the 
text as a twisting around the axis OX. The formula for this case is 
found in the simplest way by fixing our attention on the potential 
energy of the system. When the edges AB and CD are turned 
around OX through equal angles (p in opposite directions, two of 
the lines AC, AB, BC, BB are changed to 



V4:l' + a^sin^O ;r _ g,) = a(l _ ig, _ i g,^), 
and the two others to 

The potential energy due to the mutual action of the corpuscles 
is therefore 

The potential energy with respect to the positive sphere having 
not been altered (because each electron has remained at the distance 
r from the centre), and the kinetic energy being equal to 

the equation of motion becomes 



294 NOTES. 48 

-1- ma^a? + - — w^ = const., 



gmng for the frequency 



36' e„e 



In examining the vibrations for which the equations (181) and 
(182) are given in the text, we may treat the system as one with 
only two degrees of freedom, the configuration of which is whoUy 
determined by the coordinates ]) and g. 

This time we shall apply the general theory of a vibrating 
system, starting from the formulae for the potential energy JJ and 
the kinetic energy T expressed as functions of p,g,p,9- If we 
ascribe a potential energy zero to two corpuscles placed at the distance 
a, their potential energy at the distance a -\- da will be 

e* e^ c' i Sa {3a) 



in{a -\-Sa) ina 45i 



a* ^" a' r 



The value of da being 2g for the pair AB, — 2g for CD, and 
— for the remaining pairs, we find the following expression for the 
mutual potential energy of the four corpuscles 






9'--ieQ,g' m 



As to the potential energy m of a corpuscle with respect to the 
positive sphere, we may write for it e{(p — qpj, if the potential due 
to the sphere has the value (p^ for the position of equilibrium of the 
corpuscle and the value qp for its new position. Therefore, since (p 
is a function of the distance r from the centre, we may write, deno- 
ting by (Jr the change of r, 

Takmg into account that, by Poisson's equation, 

d^qp 2 d()p 

dr°' r dr ^' 

and that — ^ , the electric force actiug on the electron in its ori- 
ginal position, is equal to 

e J^ 

4«P ~ y Po»") 
we find 



48 NOTES. 295 

li A'B' is the line AB displaced, and E' its middle point, we have 

OE' = l+p, E'A' = i,a + g, 

OA' =Y{f+p)^+(Xa + cjy, 
and therefore for the electron A 

The same value holds for B and we get those for G and D by 
changing the signs of p and g. Substituting in (90) and taking the 
sum of the four values, we find 



which, added to (89), gives 

f^ = e [ -2 (Po - q) ~. (2 Ip + agf-^ q, (2p' + 'ig') 
= 2m(ccif' + 2^pg + yg^), 



if we put 






The square of the velocity heing p^+g^ for each electron, we have 
T=2m{p' + g'), (91) 

and the equations of motion 



dt\dp)^dp ' dt\d'g) ^ dg ~ 

assume the form 

p + ap + ^g = 0, g + ^p-\.y(j = Q. 

If we put 

p = Jc Goant, g = sp, 

the constants n and s are determined by the equations 
-w2 + (a + i3s) = 0, -sn2 + (/3 + ys) = 

from which (181) and (183) are easily deduced. 

In the calculation of the influence of a magnetic field on the vibra- 
tions to which the formula (183j relates, we may consider the three 
modes of motion, corresponding to a definite value of s, which, in the 
absence of a magnetic field, have the same frequency, say »(,, as the only 
ones of which the system is capable. Reverting to the formulae of § 90, 



296 NOTES. 48 

we shall call p^, p^, p^ the three displacements, common to all the 
electrons, which occur in the three modes, this displacement being 
parallel to OX in the first mode, to OT in the second and io OZ 
in the third. It is to be understood that p.^ is now what is called 
p in § 100, and that in every case the displacements p are attended 
with transverse displacements g = + sp. 

The equation {91) gives for each mode 

so that the coefficients m^, m^, m^ introduced in § 89 have the 
common value 

mo = 4/M(l+s^). (92) 

The coefficients f^, f^, f«, are also equal to each other, and if we 
substitute 

(cf. (175)), we find the following equations 

- iihc^ili + Wo W - »*^) ^2 - i^Ciili = 0; ■ (93) 

- inaC^iQi - i^Cas 1^ + ^o (V — '^^) ii=^,. 

corresponding to (176) and giving for the frequencies of the magnetic 
triplet (cf. (177)) 

n, and n^ ± g^Z-zpy^^ V'cl^cli + c\^ ■ (94) 

It remains to determine the coefficients c, for which purpose we 
have to return to (173). 

The expression P^Sp^ represents the work done, in the case of 
the virtual displacement 8p^, by the electromagnetic forces that are 
called into play by the motion of the electrons in the magnetic field H. 
Consequently, c^^ix^dp^ is the work of these forces in so far as they 
are due to the velocities of the particles in the motion determined 
by Pg. Calculating this work, we shall find the value of Cjg- 

It will be weU to introduce the rectilinear coordinates of the 
four corpuscles in their positions of equilibrium. If the axes are 
properly chosen, these are for A: I, I, I, for B: I, — Z, —I, for C: 
— I, I, —I, and for I): —I, —I, I. 

When the coordinate p^ is changed by 8p^, the four particles 
undergo a displacement equal to Sp^ in the direction of OX, com- 
bined with displacements sdp^ directed along the line AB foi A 
und B and along CB for G and B. Taking into acount that in 
the case of a positive s8p^, the distance from OX is increased for 



48 



NOTES. 



297 



for A: 


SPi, 


s'^lh, 


s'di),. 


„ B: 


^Pi, 


-s'dp^, 


-s'dp,, 


„ c 


^Pi, 


-s'dp,, 


s'dp,, 


„ D 


SPi, 


s'dp,, 


-s'dp,. 



A and B, and diminislied for C and D, and putting s' = s]/-|-, we 
find for the rectangular components of the displacement 



(95) 



If here, instead of dp,, we wrote p,, we should get the com- 
ponents of the velocities occurring in the motion p,. Similarly, the 
velocities in the motion p^ are 



(96) 



We have now to fix our attention on the electromagnetic forces 
due to these velocities, and to determine the work of these forces 
corresponding to the displacements (95). The result is found to de- 
pend on the component H^ only, and we shall therefore omit from 
the beginning aH terms with H^ and H . Thus we write 



for A 


s'ih, 


Pi, 


s'h, 


„ B 


-s'iK, 


ih, 


■^'I>i, 


„ c 


— s'ih> 


k, 


-s'i>2, 


V B 


s'i>2> 


h, 


-s'A, 



v.H. 



■v.H„ 



for the components of the electromagnetic force acting on an electron, 
by which, taking V^ and V from {96), we find the following forces 
acting on the coi"puscles in the directions of OX and OY: 



for A: 
„ B: 
, C: 
„ B: 






fill / • 

--^".sp^, 

« u '• 

-j".sp^, 

— n.sjh, 
c • -^ -' 

e li , ■ 
-^^.sp,. 



Finally, in order to find the work fi2i>2^i'i; '^e must take the 
products of these quantities and the corresponding ones in the first 
two columns of (95), and add the results. This leads to the value 



, = 44HXl-s'^) = 4--^H,(l-iA 



and similarly 



298 NOTES. 49, 50 

c.a = 4 1 H,(l - -\s'), c,,= 4.^ H,(l - -\s% 

so that the last term of (94) is equal to 

_i_ _i _ I H I •*• "^^ 
— 2cm II' 1+s^ ■ 

DiTiding this by the corresponding term in (164), we find 

_ 1— |s^ 



from which the values (184) and (185) are easily deduced. 



{97) 



49 (Page 123). Let p — (j^ be made to approach the limit 
from the positive side, so that, by (182), v = + oo. Taking into 
account that 

and that the limit of 

V 

is 2, one will easily find that the formulae (183) — (185) lead to the 
values given in the text. 

The same results are also obtained when Q — Qq is supposed to 
approach the limit from the negative side. 

It must, however, be noticed that, as (97) shows, for one of the 
two solutions (namely for the one for which m = — \) the coefficient 
s determined by (181) becomes infinite, indicating that for this so- 
lution p = Q (since g must be finite). The corresponding vibrations 
would therefore be ineffective in the limiting case (§ 99), because 
the radiation is due to the vibrations of the electrons in the di- 
rection of OX. 

50 (Page 123). After having found the frequency n, we may 
deduce from the equations [93) the ratios between q^, q^, q^, which 
determine the form of the vibrations, and the nature of the light 
emitted. We shall abbreviate by putting 

4|(l-is^) = <J, i.98) 

so that 

and, by (94) and {92), for the outer lines of the triplet 



«„c 



"0 



+ riH!, {99) 



m. 



where we shall understand by ] H | a positive number. 



50 NOTES. 299 

If h is a unit vector in the direction of the magnetic force, the 
equations (93) assume the form 

±Q2 + i{K(ls-KQi)-Q,\ {100) 

Let 

2i = a^+;b,, <?2 = a^ + ibj„ ^3 = a, + ^b, 

be a set of complex values satisfying these conditions and let us con- 
sider a^, b^, etc. as the components of certain vectors a and b. 

Separating the real and the imaginary parts of {100), we find 
the equations 

+ a-[b-h] = 0,, +b + [a-h] = 0, 

showing in the first place that the vectors a and b must be at right 
angles both to the magnetic field and to each other, and in the 
second place that they must be of equal magnitude. 

We are now in a position to determine the nature of the light 
emitted by the vibrating system. As we found in § 39, the radia- 
tion of an electron depends on its acceleration only. We infer 
from this that, when there are a certain number of equal electrons, 
the resultant radiation will be the same as if we had a single corpuscle 
with the same charge, whose displacement from its position of equi- 
librium were at every instant equal to the resultant of the displace- 
ments of the individual electrons. Now, in the first mode of motion 
which we have considered in what precedes, the resultant displace- 
ment is obviously 4jj in the direction of OX. In this way it is 
seen that the radiation going forth fi'om the tetrahedron when it 
vibrates in the manner we have now been examining is equal to that 
from a single electron, the „equivalent'' electron as we may call it, 
the components of whose displacement are given by the real parts 
of the expressions 

A:q^E'"', Aq^j*"', ^tp^e'"', 
i. e. by 

4a^ cos ut — 4 b^ sin w if, | 

Adi,j cos nt — 4bj^ sinn^, | {^^l) 

4aj cos nt — 4b^ sinnt. I 

The equivalent electron therefore has a motion compounded of 
two rectilinear vibrations in the directions of the vectors a and b, 
with equal amplitudes 4 a| and 4|b| and with a difference of phase of 
a quarter period. Hence, it moves with constant velocity in a circle 
whose plane is perpendicular to the magnetic force, and the radiation 



300 NOTES, 51 

will be much the same as in the elementary theory of the Zee- 
man- effect. 

When we take the upper signs in our formulae we have 

a = [bh], 

from which it follows that the circular motion represented by (lOT) 
has the direction of that of the hands of a clock, if the observer is 
placed on the side towards which the lines of force are directed. 
Therefore in this case the light emitted in the direction of the lines 
of force has a right-handed circular polarization. Its polarization is 
left-handed when we take the under signs. 

Now, the equation {99) shows that, when 6 is positive, the fre- 
quency is greatest for the right-handed, and least for the left-handed 
circular polarization, contrary to what we found in the elementary 
theory of the Zeeman- effect. The reverse, however, will be the 
case, when 6 has a negative value. Since the charge e is negative, 
it follows from {97) and {98) that the signs of 6 and co are 
opposite. The sign of the Zeeman-effect will therefore be that which 
we found in the elementary theory or the reverse according as to is 
positive or negative. 

51 (Page 126). When the particle has a velocity of translation V, 
the forces acting on one of its electrons are 

Here, denoting by x, y, z the coordinates of the electron with 
respect to the centre of the particle, and distinguishing by the index 
the values at that point, we may replace H^, H, , H^ by 

Ho.+ ^|"/ + 2/f^ + 4J, etc. {102) 

Substituting this in the expressions 

2:{yZ-zY), etc. 

for the components of the resultant couple and using the equations 
of § 104, we find 



^ t ^^^- 'y'-^^f - V. t ^^^+ ^'^ t ^^1' ^*- 



cy "*" dz dx ' 

c l^': dx ^ V dy ^ • dz i> ^^^- 



51 NOTES. 301 

When the field is constant and if in the symbol -77 we under- 
stand by H the magnetic force at the point occupied by the particle 
at the time t, the couple is given by 

_cKdM 

c dt ' 

and, since the moment of inertia is "ImK, the change of the angular 
velocity k is determined by 

rfk _ _ e dH 
dt 2mc dt 

Hence, on the assumption that the particle -did not rotate so long 
as it w^as outside the field, 

k = --^-H. 

2mc 

In the above calculation no attention has been paid to the elec- 
tromagnetic forces called into play by the rotation itself. In as much 
as the magnetic field may be considered as homogeneous throughout 
the extent of the particle, these forces produce no resultant couple, 
just because the axis of rotation is parallel to the lines of force. 
This is seen as follows. If r denotes the vector drawn from the 
centre to one of the electrons, we have for the linear velocity of 
that corpuscle 

v = [k-rj, 

and for the electromagnetic force acting on it 

F = |[v-H] = |{(k-H)r-(r.H)k}. 

The moment of this force with respect to the centre is 

[r.F] = -|(r.H)[r.k], 

so that its components are 

-±ixW^ + yW,^ + ,W^){y\,-,V,), etc. {103) 

From this we find for the components of the resultant moment 

--^^(H,k,-H,k^), etc., 

from which it is seen that this moment is zero when k has the 
direction of H. 

The problem is more complicated when we take into account 
the small variations of the magnetic field from one point of the par- 
ticle to another. I shall observe only that, if we use the values 



302 NOTES. 52, 53 

{102), we must add to {103) terms of the third order with respect to 
X, y, s, and that the sum of these terms vanishes in many cases, for 
instance when, corresponding to each electron with coordinates x, y, z, 
there is another with the coordinates — x, — y, — s. 

52 (Page 126). Let k and r have the same meaniag as in the 
preceding Note and let V be the absolute velocity of an electron, v' 
its relative velocity with respect to the rotating particle, so that 

V = [k ■ r] + v'. {lOA) 

From this we find for the acceleration 

q = V = [k f] + v' = [k V] + v'- 

The change of v' consists of two parts 
v'=[kv'] + q', 

where the second is the relative acceleration and the first the change 
that would be produced in v' if there were no such acceleration; in 
this case v' would simply turn round with the particle. Since, on 
account of {104^, we may write 

[k.v']=[k.v], 

when we neglect the square of k, we are led to the formula 

q = q' + 2[kv]- 

53 (Page 135). In this statement it has been tacitly assumed 
that the bounding surface G of the spherical space S does not inter- 
sect any particles. Suppose, for instance, the molecules to be so 
polarized hat each has a positive electron on the right and a negative 
one on the left-hand side, and draw the axis OX towards the first 
side. Then, when the surface e passes in all its parts through the space 
between the particles, the integral fgxdS will be equal to the sum 

of the electric moments of the particles enclosed, and may with 
propriety be called the moment of the part of the body within the 
surface (cf. equation (195)). If, on the contrary, molecules are inter- 
sected, the value of the integral does not merely depend on the com- 
plete particles lying in the space S, but it must be taken into account 
that, in addition to these, 6 encloses a certain number of negative 
electrons on the right-hand side, and a certain number of positive elec- 
trons on the opposite side. Even when these additional electrons are 
much less numerous than those belonging to the complete particles 
they may contribute an appreciable part to the integral, because the 
difference between the values of x for the positive and the negative 



64 NOTES. 303 

ones is comparable with the dimensions of the space S itself, and 
therefore much greater than the corresponding difference for two elec- 
trons lying in the same particle. 

The following remarks may, however, serve to remove all doubts 
as to the validity of the relation 

pV = P. 

When the molecules are irregularly arranged, as they are in li- 
quids and gases, some of them (and even some electrons) are cer- 
tainly intersected by the spherical surface used in the definition of 
the mean values 9. But, on account of the assumptions made about 
the dimensions of 6, the intersections wiU be much less numerous 
than the molecules whoUy lying within the surface, and if, in calculating 
j(pdS, we omit the parts of particles enclosed by 0, this will lead to 
no error, provided that the function qp be of such a kind that the 
contribution to the integral from one of those parts is not very much 
greater than the contribution from one of the complete particles. 

This condition is fulfilled in the case of the integral J()\^dS, 
because there is no reason why the velocities V should be exceptio- 
nally great near the surface 0. Without changing the value of the 
integral, we may therefore make the surface pass between the par- 
ticles (by slightly deforming it), and then we may be sure that 

f Q\^dS= J- I QxdS, and that the latter integral represents the total 
electric moment of all the complete particles in the space S. 

54 (Page 138). We shall observe in the first place that the 
field in the immediate neighbourhood of a polarized particle may be 
determined by the rules of electrostatics, even when the electric mo- 
ment is not constant. Take, for instance, the case treated in § 43. 
It was stated in Note 23 that at great distances the terms resulting 
from the differentiation of the goniometric function are very much 

greater than those which arise from the differentiation of y ■ These 

latter, on the contrary, predominate when we confine ourselves to 
distances that are very small in comparison with the wave-length; 
then (cf. (88) and (89)) we may write 

a = 0, 
d = — grad qp, h = 0, 
from which it appears that the field is identical with the electro- 
static field that would exist, if the moment p were kept constant. 



304 NOTES. 54 

It is further to be noted that the difference between the mean 
electric force E and the electric force existing in a small cavity depends 
only on actions going on at very short distances, so that we may 
deal with this difference as if we had to do with an electrostatic 
system. 

Let us therefore consider a system of molecules with invariable 
electric moments and go into some details concerning the electric 
force existing in it. 

The field produced by the electrons being determined by 

Agj = — p, 
d = — grad q), 

we have for the mean values 

A<p = — (0, 
E = d = — grad 9 , 

or, in words: the mean electric force is equal to the force that 
would be produced by a charge distributed with the mean or, let us 
say, the „effective" density q. 

In the definition of a mean value 9 given in § 113, it was ex- 
pressly stated that the space S was to be of spherical form. It is 
easily seen, however, that we may as well give it any shape we like, 
provided that it be infinitely small in the physical sense. The equation 

may therefore be interpreted by saying that for any space of the 
said kind the effective charge (meaning by these words the product 
of Q and S) is equal to the total real charge. 

We shall now examine the distribution of the effective charge. 
Suppose, for the sake of simplicity, that a molecule contains two 
electrons situated at the points A and J3 with charges — e and + e, 
and denote by r the vector AB. There will be as many of these 
vectors, of different directions and lengths, as there are molecules. Now, 
if the length of these vectors is very much greater than the size of 
the electrons, we may neglect the intersections of the bounding sur- 
face of the space S with the electrons themselves, but there will be 
a great number of intersections with the lines AIB. These may not 
be left out of account, because for any complete molecule fgdS = 0, 
whereas each of the said intersections contributes to the effective 
charge within an amount — e or -(- e according as r^ (where n is 
the normal to drawn outwards) is positive or negative (cf Note 53). 



54 NOTES. 305 

Hence, the total charge within 6 may be represented by a surface 
integral. In order to find the part of it corresponding to an ele- 
ment d0 (infinitely small in a physical sense) we begin by fixing our 
attention on those among the lines AB which have some definite 
direction and some definite length. If the starting points A are ir- 
regularly distributed and if, for the group considered, their number 
per unit of volume is r, the number of intersections with ds will be 
vr^de when r„ is positive, and — vr^ds when it is negative. There- 
fore, the part contributed to the charge within (j is — ver^de in 
both cases, and the total part associated with ds is —ZveC^de, the sum 
being extended to all the groups of lines AB. But er is the electric 
moment of a particle, ver the moment per unit of volume of the 
chosen group, and —vcr the total moment per unit of volume. De- 
noting this by P, we have for the above expression — Ever^de the 
value — P^, and for the effective charge enclosed by the surface 6 

-fP„d6. {105) 

As the difference between E and the electric force in a cavity 
depends exclusively on the state of the system in the immediate vici- 
nity of the point considered, we may now conceive the polarization P 
to be uniform. In this case the integral (105) is zero for any closed 
surface entirely lying within the body, so that the effective charge 
may be said to have its seat on the bounding surface E. Its surface 
density is found by calculating (105) for the surface of a flat cy- 
linder, the two plane sides of which are on both sides of an element 
dZ a.i a distance from each other that is infinitely small in compa- 
rison with the dimensions of dE. Calling N the normal to the sur- 
face E, we have at the "outer plane P„ = (if we suppose the body 
to be surrounded by ether), and at the inner one P„ = — Pjf. The 
amount of the effective charge contained in the cylinder is therefore 
CTiven by PydZ, and the charge may be said to be distributed over 
the surface with a density Pjv. 

Now consider a point A of the body. By what has been said, 
the electric force E at this point is due to the charge P,v on the 
bounding surface E. If, however, a spherical cavity is made around 
A as centre, there wiU be at this point an additional electric force 
E', caused by a similar charge on the walls of the cavity, and ob- 
viously having the direction of P The magnitude of this force is 
found as follows. Let a be the radius of the sphere, de an element 
of its surface, d' the angle between the radius drawn towards this 
element and the polarization P. The surface density on de being 
— i P I cos -a-, we have for the force produced at A 

\P\ C0B^»d6, 






L r e n t z , Theory of electrons. "0 



306 NOTES. 55-67 

giving 

E'=^P. 

Our foregoing remarks show that the exf)ressioii 

E + iP 

may always be used for the electric force at the centre of a sphe- 
rical cavity, even though the polarization of the body change from 
point to point and from one instant to the next. 

55 (Page 138). In the case of a cubical arrangement all the 
particles within the sphere may be said to have equal electric mo- 
ments p. Taking the centre A of the sphere as origin of coordinates, 
we have for the force exerted in the direction of a; by a particle si- 
tuated at the point {x, y, s), at a distance r from the centre, 

Pj. Sx^ — r" pj, Sxy p. Sxz 
But the sums 



^ p ' ^ 7.6" ) ^ -fir 



are zero, when extended to all the particles within the sphere. For 
the second and the third sum this is immediately clear if we take 
the axes of coordinates parallel to the principal directions of the 
cubical arrangement. Further, for axes of this direction, 

showing that each of these expressions must be zero, because their 
sura is so. 

56 (Page 139). It must be noticed that this magnetic force H 
produces a force 

4 [V • H] 



acting on an electron. Since, in a beam of light, H is in general of 
the same order of magnitude as the electric force E (cf. the equations (7)), 

this force is of the order of magnitude — in comparison with the 

force eE. It may therefore be neglected because the amplitudes of 
the electrons are extremely small with respect to the wave-length, 
so that the velocity of vibration is m\ich smaller than the speed 
of light. 

57 (Page 141). In order to compare the effect of the collisions 
with that of a resistance of the kind represented by (197), we shall 



67 NOTES. 307 

first, consider the vibrations set up in an isolated particle whose 
electron is subjected to a periodic electric force 

E^ =p cos nt {106) 

and to the forces determined by (196) and (197j. The equation 
of motion 

is most easily solved if, following the method indicated in § 119, we 
replace (106) by 

In this way we find for the forced vibrations 

i = ^ gint ^ P^ int (10?) 

f — mn''-{-ing m(»?„* — n^)-\-ing ' \ ') 

where 

v = -- 

" m 

Let us next suppose that there is no true resistance, but that 
the vibrations of the electrons are over and over again, disturbed by 
impacts occurring at irregular intervals. In this ease the motion of 
each particle from the last collision up to the instant t for which we 
wish to calculate ^, is determined by the equation 

the general solution of which is 

where the integration constants C^ and C2 will vary from one par- 
ticle to another. These constants are determined by the values of | 

and -jT, say (^)q and I-tt) , immediately after the last collision. Now 

among the great number of particles, we may distinguish a group, 
still very numerous, for which the last collision has taken place at a 
definite instant t^. Supposing that, after the impact, all directions of 
the displacement and the velocity are equally probable, we shall find 
the mean value of § for this group, if in (108) we determine C^ and 

G2 by the conditions that for t = t^ both | and , vanish. The 

result is 

* m(M„*— M*)l 2 \ nj 2 \ nj j ' 

or, if we put 

20* 



308 NOTES. 58 

This is the mean value of | taken for a definite instant t and for 
those particles for which a time ■9- has elapsed since their last col- 
lision, and we shall obtain an expression that may be compared 
with (107), if we take the mean of (109) for all the groups of par- 
ticles which differ from each other by the length of the interval &. 
Let N be the total number of particles considered, and A the 
number of collisions which they undergo per unit of time, so that 
the time r mentioned in the text is given by 

N 

The collisions succeeding each other quite irregularly, we may reckon 
that the number of the particles for which the interval &■ lies 
between & and & -\- d& is 

As ^dd- = ^e "^dd-; (110) 

this is found by a reasoning similar to that which we used in Note 36. 
We must therefore multiply (109) by (110), integi*ate from d- =0 
to # = oo, and divide by N. In this way we get for the final mean 
value of the displacement 






Neglecting the term -, in the denominator we see that under the 

influence of the collisions the phenomena will be the same, as if there 
were a resistance determined by 

2m 



58 (Page 147). In the case of a mixture the electric moment 
P is made up of as many parts P^, Pg, . . . as there are constituents. 
Reasoning as in §§ 116 — 119 we can establish for each component 
formulae like (200), so that, if we put a = J-, we have for the first 
substance 



for the second 






59, 60 NOTES. 309 

and so on. Hence, if all the dependent variables vary as a*"*, 

D ■■ . E + jP p _ E + ^P 

fY—m^'n'" 2 fY — J»j'n*'''*' 



and, if we put 

1 , 1 __ , _ ,, 

Combining this with (192) we find 

n = ^-ii " F 

and for the index of refraction 

Now each term --j-^ , -.t siTes the value of ^, , „ for one of 

3(/ — m n^) ^ (<■ + 2 

the constituents taken with the density mq which it has in the 
mixture, a value that is found when we multiply the constant r for the 
constituent in question by the density niQ. This immediately leads 
to equation (218). 

59 (Page 149). According to the equations (220), if we put 
a = ^-, the displacements |j, Sj; •• • ^re determined by 

(/;-m,«2)li = e,(E, + iPJ, etc. 

Consequently 

with similar formulae for Ne^T]^, Ne-^^^, etc. Hence, taking the sums, 
P (f \ ^ P\i ^"'^ I ^^^ ' I 1 

from which the formula (222) is easily found. 

60 (Page 153). The direct result of the substitution is 

/c* cVc^\ _ . ic^k _ . 1 _ . a — i§ 

\v' n' ) *Tw ~ ^" a + i/3 ^ '^ a' + §'' 

giving 

,l2 _ i- - =1-1- -^ 

from which the equations (227) and (228) are easily deduced. 



310 NOTES. 61, 62 

61 (Page 154). The expression -rT"fl2 considered as a function 

of a has a maximum value -^ for a = /3: it is therefore rery small 
when j3 is large. It follows from this that even the greatest values 
of -^Tn?2 ^^® ^^ *^^ order of magnitude -j, so that we may ex- 
pand the square root in (227) and (228) in ascending powers of 
that quantity. Hence, if we neglect terms of the order ^3 , 



V 



. 2« + l _ -, , 1 2a + 1 1 (2« + l)- 

~^ „2 I 02 -"-In 



a2 + ^2 - 1 2 a'+ ^^ 8 (a2 + 

and this may be replaced by 

. Jt_ 2o:+ 1 1 a^ 



2 a* + /32 2 (a2_j_(32)!!> 

because the quantity 7 tjT 52V2 never has a value greater than one of 

the order -^ ■ 
Finally 



We are therefore led to (229) and (230) if in fi we neglect terms of 
the order -^, and in h terms of the order ^^. Indeed, if we want 

to know h with this degree of approximation, we may omit in Ic^ and 

c^h^ 1 

in — Y quantities of the order -^, as we have done in (HI). 

62 (Page 156). If Jdn is the intensity of the incident light, 
in so far as it belongs to frequencies between n and n -{- dn, the 
amount of light that is absorbed by a layer with the thickness z/, 
upon which the rays fall in the direction of the normal is given by 
the integral 

^=/(l-6-2*^)J'rfw, 

where we have taken into account that the intensity is proportional 
to the square of the amplitude. If the absorption band is rather 
narrow, we may put 

7, = < . _J. 

2c c<:« + |32 

and, in virtue of (231), 

dn = — ~~, — -, da ■ 
2 m w„ 



6S 64, NOTES. 311 

Further, we may extend the integration from a = — c<j to « = + oo, 
considering j3 = n^' g' and J as constants. The calculation is easily 
performed for a thin layer, for which 

1 - £-2*^ = 2kzl - 2F^^. 

It is found that the part of A that is due to the first term is 
independent of g' or g. When, however, the second term is retained, 
A increases with the resistance g. 

63 (Page 161). This is easily found if the denominator of (239) 
is written in the form 

{«(l+«)-^2-y^) + /(l+2«)/3 

and then multiplied hy the conjugate complex expression. 

64 (Page 167). The explanation of magneto-optical phenomena 
becomes much easier if the particles of a luminous or an absorbing 
body are supposed to take a definite orientation under the action of 
a magnetic field. On this assumption, which makes it possible to 
dismiss the condition of isotropy of the particles (§ 93), Voigt') has 
been able to account for many of the more complicated forms of 
the Zeeman-effect; it was found sufficient to suppose that each par- 
ticle contains two or more mutually connected electrons whose motion 
is determined by equations similar to our formulae of § 90, the recti- 
linear coordinates of the electrons now taking the place of the general 
coordinates p. The theory thus obtained must undoubtedly be con- 
sidered as the best we possess at present, though the nature of the 
connexions remains in the dark, and though Voigt does not attempt 
to show in what manner the actions determined by the coefficients c 
are produced by the magnetic field. 

I must also mention the beautiful phenomena that have been 
discovered by J. Becquerel. ^) Certain crystals containing the ele- 
ments erbium and didymium show a great number of absorption 
bands, many of which are so sharp, especially at the low tempera- 
tures obtainable by means of liquid air or liquid hydrogen^), that 
they may be compared with the lines of gaseous bodies, and these 
bands show in remarkable diversity the Zeeman-effect and the 
phenomena connected with it. Of course, in the case of these crystals 



1) W. Voigt, Magneto- und Elektrooptik, Leipzig, 1908. 

2) J. Becquerel, Comptes rendus 142 (1906), p. 776, 874, 1144; 143(1906), 
p. 769, 890, 962, 1133; 144 (1907), p. 132, 420, 682, 1032, 1330. 

3) H. Kamei-lingli Onnes and J. Becquerel, Amsterdam Proceedings! 
10 (1908), p. 592. 



312 NOTES. «5 

the hypothesis of isotropic particles would be whoUy misplaced. Voigt 
and Becquerel found it possible to explain the larger part of the 
observed phenomena on the lines of Voigt's new theory to which 
I have just alluded. 

In § 91 it was stated that a true magnetic division of a spectral 
line is to be expected only when the original line is in reality a 
multiple one, i. e. when, in the absence of a magnetic field, there are 
two or more equal frequencies. Voigt has pointed out that, when 
originally there are two frequencies, not exactly but only nearly 
equal, similar effects may occur, sometimes with the peculiarity that 
there is a dissymmetry, more or less marked, in the arrangement of the 
components observed under the action of a magnetic field. Cases of 
this kind frequently occur in Becquerel's experiments, and Voigt 
is of opinion that many of the dissymmetries observed with isotropic 
bodies (§ 142), if not all, may be traced to a similar cause. 

It is very interesting that some of Becquerel's lines show the 
Zeeman-effeet in a direction opposite to the ordinary one (i. e. with 
a reversal of the circular polarization commonly observed in the lon- 
gitudinal effect) and to a degree that is equal or even superior to the 
intensity of the effect in previously observed cases. These phenomena and 
similar ones occurring with certain lines of gaseous bodies^) have led 
some physicists to admit the existence of vibrating positive electrons, for 

which the value of — would be comparable with or even greater than 

the value found for the negative electrons of the cathode rays. Thej"^ 
may also be explained by the assumption that in some systems of 
molecules, under the influence of an external magnetic field, there are 
motions of electricity such as to produce in the interior of the par- 
ticles a field that is opposite to the external one. To this latter hypothesis 
Becquerel, however, objects that, like all phenomena of induced 
magnetization, the internal fields in question would in all probability 
be liable to considerable changes when the body is heated or cooled, 
whereas the magnetic division of spectral lines remains constant 
through a wide range of temperatures. 

The possibility of a third explanation, though one about which I 
am very doubtful, is perhaps suggested by what we found in § 102, 
namely by the reversion of the ordinary direction of the effect caused 
by a particular arrangement of a number of negative electrons. 

65 (Page 171). If x, y, z are the coordinates of a particle of 
tbe medium at the time t, its coordinates at the time t -^ dt will 
be equal to 



1) J. Becquerel, Comptes rendus 14G (1908), p. 683; A. Dufour, ibidem, 
p. 118, 229, 634, 810; R. W. Wood, Phil. Mag. (6) 15 (1908), p. 274. 



66, 67 NOTES. 313 

a=' = ^ + Sa^dt, y' = y + Q^^dt, n' = z + g^dt. 

Here g^, g^,, g^ may be regarded as linear functions of x, y, s, 
so that, for instance, 

9x = « + /3a; + 72/ + Sz, 
or, as we may write as well 

9x = a + /5x' + yy' + 6/. 
The particles which originally lie in the plane 

X = a 
will have reached the plane 

re' = a + (« + /3a;' + yij + dz) dt 

at the end of the interval considered. The direction constants of the 
normal to this plane are proportional to 

1-fidt, ~ydt, -6dt 



or to 



1_|8^^<, -q^-dt, ~^B- dt. 



66 (Page 173). Let a sphere of radius B move with the con- 
stant velocity iv through an incompressible medium, and let us sup- 
pose the motion of the latter to be irrotational. Then, if the centre 
of the sphere is taken as origin of coordinates, and the line of mo- 
tion as axis of x, the velocity potential is given by 

giving for the components of the velocity 

^ _ Jl P3 ,,. 3 a;' — ?■' 



dx 

At a point of the intersection of the surface with the plane YOZ, 
these values become 

-i-w, 0, 0, 

so that the relative velocity of sliding is — -|w. 

67 (Page 173). Instead of considering a uniform translation of 
the earth through the ether, we may as well conceive the planet to 
be at rest, and the ether to flow along it, so that, at infinite di- 
stance, it has a constant velocity w^ in the direction of OZ. 



314 NOTES. 67 

Let the ether obey Boyle's law, and let it be attracted by the 
earth with a force inversely proportional to the square of the distance 
r from the centre. Then, when there is no motion of the medium, 
the density k and the pressure p wiU be functions of r, determined 
by the equation of equilibrium 

dp coJc 

dr r^ ' 
and the relation 

Ic = ^p, 
where m und ^ are constants. 

These conditions are satisfied by 

log/« = -^ + log/(;o, 

JvQ beiag the density at infinite distance. 

Now, there can be a state of motion in which there is a velo- 
city potential (p, and in which the density Ic has the value given by 
the above formula. Indeed, if we put 



(p = z 



«6v-i)+K'^+i)^'; 



(understanding by a and & constants and taking the centre of the 
sphere as origin of coordinates), the components of the velocity 



dw Of dw 

dx' oy' dz 

satisfy the equation of continuity 

8{lcu) d(kv) d(kw) __ „ 
dx dy dz 

The form of <p has been chosen with a view to the remaining 



conditions of the problem, namely: 

dx ' dy ' dz 



for r=oo: — = 0, ~ = 0, -^ 



and 

for r = R (i. e. at the surface of the earth): -^ 

These conditions lead to the equations 



-(S+^;+i)'"''»- 



Along the intersection of the planet's surface with the a;i/-plane 
there is a velocity of sliding 

_ /i 0) /i m 



«8 NOTES. 315 

This is found to be 0,011 Wo if ^ = 10, and 0,0056 »„ if ^ = 11. 

In these cases the ratio between the density near the surface and 
that at infinite distance would be s^° or s^^ respectively. 

68 (Page 181). Let the relative rays converge towards a point 0, 
which we take as origin of coordinates, and let us determine the form 
of the waves by the construction explained in § 153. We have to 
compound a vector in the direction of the relative ray and having 

the magnitude v' with a vector — ,. Neglecting quantities of the 

second order, we may also make the iirst vector equal to v, the wave 
velocity in the medium when at rest, and we may consider this ve- 
locity as constant in the immediate neighbourhood of the point 0. 
Moreover, the second vector may be regarded as having a constant 
magnitude, say in the direction of OX. 

Now, at a point (a;, y, z), at a distance r from 0, the compo- 
nents of the first vector are 

X y z 

r ' r r ' 

and those of the second 

-ifll 

so that the components of the resultant vector, which is at right 
angles to the wave-front, are 



f a; , g h y z 

\ r ' u^ J ' r ' r 



The equation of the surface normal to the resultant vector is 
therefore 

vr -\-^X;X= G, 

This is the equation of an ellipsoid, the centre of which has 
the coordinates 

__5^ 

if 

ft ' 

and whose semi-axes have the directions of OX, OF, OZ and the 

lengths 

vC C C 



V^—CC"' yt,2_a8' |/„2_^8 



316 NOTES. 6» 

Since the square of a is neglected, we may say that the waves 
are of spherical form. Their centre approaches the point as the 
constant C diminishes. 

69 (Page 191). If n is the frequency of the source of light, 
the frequency at a fixed point in one of the tubes will also be n, 
because the successive waves take equal times to reach this point. 
Hence, with reference to fixed axes, a beam of light may be repre- 
sented by expressions of the form 



a cos wu \-p), 



where u is the velocity in question. 

Transforming to axes moving with the fluid — and confining 
ourselves to one of the two cases distinguished in the text — , we 
have to put 

X = x' -{- wt, 

by which the above expression becomes 

a cos nit t Vv]- 

In this way the relative frequency is seen to be 

for which, denoting by fi the refractive index for the frequency n, 
we may write 

c 
because u differs from — only by a quantity proportional to w. 

The index of refraction corresponding to the frequency n is 

iiiv du, 
' c an ' 

and the corresponding velocity of propagation 

c c w dfi, c IV rpdfi c IV d (I 

iiw du, u, u, dn ii u, dT u, u, dX' 
c dn 

if /I is the wave-length. 

This is the velocity to which we must add the term w(l A • 

In the case of water we have for the spectral line D 

1- A = 0,438 
and 

1--,- -llf = 0,451 



70-72 NOTES. 317 

whereas, if the velocity relative to the fixed parts of the apparatus 
is represented by 

(I — 

e = 0,434 (with a possible error of + 0,02) is the value which 
Michelson and Morley deduced from their experiments. 

70 (Page 191). For the case of a mirror the proposition is 
easily proved after the manner indicated in § 154. If, supposing the 
mirror to be made of a metallic substance, we want to deduce the 
same result from the theorem of corresponding states (§§ 162 and 165), 
we must first extend this theorem to absorbing bodies. This can 
really be done.^) 

71 (Page 192). Beams of light consisting of parallel rays, in a 
stationary and in a moving crystal, wiU correspond to each other 
when their lateral boundary is the same, i. e. when the relative rays 
have the same direction s. In both cases we may consider a defi- 
nite line of this direction, and write down the equations for the 
disturbance of equilibrium at different points of this line, reckoning 
the distance s from a fixed point of it. For the stationary crystal 
the vibrations are represented by expressions of the form 

a cosn U \-p) , 

and the corresponding expressions for the other case have the form 

a cos n (t' h p) ; 

or, since along the line considered 



acosn{t—^-~+pj, 



from which it appears that the velocity u' of the ray relative to 
the ponderable matter is determined by 



1 1 , w, , W 

- -, = — I , u = u r w» 



72 (Page 194). Strictly speaking, it must be taken into account 
that in the moving system the relative rays may slightly deviate 

1) See H. B. A. Bockwinkel, Sur les pWnomenes du rayonnement dans 
un systeme qui se meut d'une vitease uniforme par rapport a Tether. Arch, 
neerl. (2) 14 (1908), p. 1. 



318 NOTES. 73 

from these lines, the theorem that their course is not altered by a 
translation having been proved only when we neglected terms of the 
second order. Closer examination shows, however, that no error is in- 
troduced by this circumstance.^) 

73 (Page 197). In virtue of (286) and (287) we have the 
following general relations between the differential coefficients with 
respect to x, y, z, t and those with respect to x, y, z , t' 

d 11 S 1 ■, w d d 1 d d id (iio\ 

^ = U^-hUv^- {113) 

The equation (17) therefore assumes the form 

dx ' dy dz c^ 8t ^ ' ^ ^ 

and the first of the three equations contained in (19) becomes 

i/dh, dhii\ M rd,. , , w Sd, ,1 /.. , ^ r-i-i~\ 

^ W - ^') = 'o it' - ^^ -7 W + c ^ ("- + ^)- (^^•') 

Substituting the value of -^ taken from this formula in (114), 
we find 

dx dy dz' c \dy' dz'l c- dx \ c^ ' ^ c^ "' 

where 1 ^ may be replaced by ^ . 

I 



h 
Hence, multiplying by -^ and taking into account the values 



of A'^ etc., 

gd; dA'y ad; _ , wu'^ , 

dx "•" dy'^ dz ^ c^ ^' 



which is the first of the equations (292), 

dx' 



P) ri 

If, on the other hand, the value of ^f drawn from (114) is 



substituted in {115), one finds 

7 /'ilk _ ^^y\ _ i^ ^Is'^ _ l5 ^^ _ /M _ '^"'^ 8d^ 1 
\dy' dz' ) c dy'' c dz' ~ \c c" ) dt' +7^"^' 

or , after multiplication by -4 > 

1) Lorentz, De I'influence clu mouvement de la terre sur les pli(5nomene3 
lumineux, Arch, neerl. 21 (1887), p. 169 — 172 (Abhandlungen fiber theoretisohe 
Physik, 1, p. 389—392). 



74 NOTES. 319 

8h^ _8yy _ 1^ (dd^ , , ,\ 
dy' 8z' c \dt' '^ ^"^ )' 

This is the first of the equations contained in 

The remaining formulae are obtained by similar transformations. 

74 (Page 198). The equations (292) have the same form as 
(100)— (103) and the proof of (295) and (296) is therefore almost 
identical with that given in Note 25 for the formulae (106) and 
(107), the first of which takes the form of (295) if the translation 
is supposed to have the direction of OX. Therefore, the following 
indications may suffice here. 

Since, in the equations (292), the independent variables are t' 
and the quantities x', y', s defined by (287), it is convenient to regard 
these latter as the coordinates of a point in an auxiliary space S" 
(cf § 170), and to consider the potentials qp' and a' as quantities to 
be defined in this space. Instead of determining them by means of 
the differential equations (294), we may as well assume the formulae 

and 

comparable with (104) and (105). In these formulae dS' denotes an 
element of the auxiliary space, r its distance from the point P' for 
which we want to calculate the potentials, and the square brackets 
serve to indicate that, if we wish to know tp and a' for the value i 
of the fourth independent variable, we must use the values of q! and 

p'u' corresponding to the element dS' and to the value i — — of 

that variable. 

Starting from {11(j) and (ii7), we can show (cf. Note 25) that 
qo' and a' satisfy the equations (294) and that they are connected 
by the relation 

Having got thus far, one finds by direct substitution that the 
values (295) and (296) satisfy the equations (292) (see, however. 
Note 6). 

As to the formula {118), the proof of it depends on the equation 

||: + div'(,'u')-5^'^3>') = o, {119) 



320 NOTES. 7-S 

just as, in Note 25, (44) was deduced by means of (45). It remains 
therefore to show that (119) holds true for the case now under con- 
sideration. 

In order to examine this point, we shall again start from equation 
(5) of Note 2 or 

%+wll + di.(,u)^0. 

On account of (287) and (288) we may write for this 

T -iSq 77 20,77 So 1 1 1^^ do , T idiQUj^ 

M^ - Mw ^, + hlw -..-^ — M ^ ^, + Id —^-f 

V t OX c oc c V t ex 

' dy dz c^ ot 

This leads to the equation (119), if, after having combined the two 
terms containing ^, we express q and U in p' and u', dividing at the 
same time by P. 

7S (Page 203). The exact formulae are found as foUows. If 
«■,.> Vr' ^r ^^® functions of i, the quantities 

may be considered as functions of 

i:=\t-u'^,x^. (120) 

We have the following relations between simultaneous infinitely 
small changes 

dx = kldx^, dy' = Idy^, dz = Idz^, 



dt' = T dt — Id -^ dx,. 



Hence 









dt' _ 
dt " 


I 




dx^ 
dt ' 






dx' 
dt' 


= 


hi 


dx^ 

dt 




h^ 


dx^ 
dt 




I ,. 


, 1^ 


'dx^ ~ 


L-7c' 


, ID dXj. ' 








h c^ 


—r- ] 
dt 


c« dt 


dy 


- = 


1- 


^ dt 


dx^ 
dt 


d^ 

' dt' ~ 




dz^ 
'" dt 


dt' 


1- 


w dx,. 
''' c^ dt 



(121) 



d X (in d z 

The values of ^^,, ^^>%, j-^,-^ are found by differentiating the 
last expressions with respect to t, and dividing the results by (121). 



76 NOTES. 321 

76 (Page 207). Beginning again with the field produced by a 
single electron, we may repeat almost verbally what has been said 
in Note 26. Thus, let P be the distant point, M the „active"^) 
position of the centre of the electron, t^' the local time (of M) at 
which this position is reached, r' the eifective distance between 31 
and P. Then 

r' = c(i'-V), 
or, if x'p, y'p, /^ are the eflPective coordinates of P (M being taken as 
origin), 

^'p'+y';^+^\f=c\t'~Q\ {122) 

Let us further denote by x', y, s the effective coordinates of some 
point Q of the electron at the time i^ (local time of M.), by x^, 
Vai ^a ^^ effective coordinates of the same point in its active po- 
sition Q^, and by t^ + t (local time of JM) the time at which this 
latter position is reached, so that, according to (288) (for which we 

may write t' = jt ^x) the local time of Q^ itself is 

ta=U+^-~^'>^^d- {123) 

The condition that Q^ shall be the active position of the point 
in question is 

{x^- x:y+ (/,- yy+{^;-z:r=-c\t'-t:y ^m) 

Now, {x, y, z) and [xj, y^, z^) are the effective coordinates of 
the same point of the electron at two moments for which the local 
time of jSL difi'ers by t. The difference between the corresponding values 

of i is yT, as is seen from formula {120), if in it we put x^= 0, 

as it is for the point M. But during the interval y r the relative 
coordinates of Q change by 

h k h 

J"x^, T".^, T".^, 

so that, on account of (28G), the effective coordinates of the same 
point undergo the changes 

h^ii^t, hUyX, fcu.T, 
or, in virtue of (289), 

We may therefore write 

xd=x'+u^x, y;=y'+yiyr, ^;=/+u;t. 



1) This term is chosen here because the word „effective" has now been 
used in another sense. 

Lorentz, Theory of electrons. 21 



322 NOTES. 76 

If these values, together with {123) and {122), are substituted in 
{124), and if we neglect terms of the second order with respect to 
r, x, y, z, we are led to the formula 

{x'x'p + y'y'p+ z'^'p)+ — r'x 
where U*"' is an abbreviation for 

y{xpU'^+ijpU',j+0'pu:). 

We may next consider two elements dS' and dS^' comparable 
with the elements dS and dS^, of which we spoke in Note 26. The 
ratio between their magnitudes is given by the functional determi- 
nant of x^', y^', s^ with respect to x', y, z, i. e. by 

' ^ dx ' y dy ' dz ' 

or, if we confine ourselves to terms of the first order with respect 

to u;, u;, u;, by 

We have now to use the formulae {116) and {117). In the first 
of them we must replace dS' by 

and, on account of the smallness of the dimensions of the electron 
in comparison with the distance from the point P, r may be taken 
equal to the effective distance between M and P. Consequently, 
since, by (290) and (302) q dS' = QdS, and fQdS = e, the scalar 
potential is given by 

where the meaning of the square brackets will be obvious. 
The vector potential is equal to 



6£^- («s) 



Finally, the electromagnetic field produced by the moving electron 
is given by (295) and (296), the first of these formulae taking the 
simple form 

<!^'=~^^-E^^A'{¥), {126) 



77 NOTES. 323 

if we put 

(g, ) = qp _ a^ 



or 



(^')=^[4(l + 4)]- il27) 

Let us now return to tlie comparison of two systems g and ^^q, such 
as we liave considered at the beginning of § 175, the one moving 
and the other at rest, and in which the effective coordinates of the 
separate electrons are the same functions of the effective time. It 
has been shown in § 173 that for an electron moving in f-^ the compo- 
nents of the vector u' are equal to the differential coefficients -jy , 

-X, -TTT. Therefore, the vector u' occurring in the formulae {121"), 

(125), {126) and (296j which determine the field in g is equal 
to the velocity of the electron in gg, from which we may con- 
clude that the values of {ip'), a', d' and h' in g are equal to those 
of tp, a, d, and h in §„; indeed, the formulae {127), {125), {126) 
and (296) are exactly similar to (79), (33) and (34) for this 
latter system. If we apply this to the case, considered in § 175, 
that the electrons perform small vibrations in the particles of the 
body, we see at once that the field belonging to such a particle in [^ 
must be determined by the equations (308), because we kaow that 
a particle in J^,, gives rise to a field determined by (88), (89), (33) 
and (34). 

77 (Page 211). Let g be a moving electrostatic system and 

j^Q the corresponding stationary one. We have a' = 0, h'=0, and, 

if q)' is the scalar potential in gp, the equations (291) and (295) give 
for every point of g 

and consequently 

H 72 ^^' ri — 7-72^ ri — — IP ^ 

' X ^ ' y c dz ' ' c dy 

From this we find for the first component of the flow of energy in g 

s.=<d,h.-d.h,)=/.^^^.((li;)^+g^;n, 

21* 



{128) 



324 NOTES, 78—80 

and (by (53) and (302)) for the first component of the electro- 
magnetic momentum, with which alone we are concerned, 

G.=q;^/((i)'+ m\ "« - -f.i'm^ WfV'- 

We have therefore merely to calculate the last integral for the 
field of a sphere without translation with radius i? and charge c. 
This is a very simple problem. We may observe that the three 
integrals 

have equal values, so that they are each equal to one third of their 
sum, i. e. to two thirds of the energy of the system. The latter 

having the value ^ — ^ , we have for each of the integrals j^—^ , and 

It is clear that G^ = and G^ = 0, so that in general 
G=7r^^fc^w. 

78 (Page 213). The equations {128) lead to the following value 
of the electromagnetic energy 

^'/l©r+*'(>+$)(e<) +©)!>« 
-i^/[(g)'+''('+?)i(if)+(i?)'i>^' ('^») 

Putting I = 1 and remembering that each of the integrals 

/ i-i^'A dS' etc. has the value— — ^, we find 
J \dxj 127tB' 

which becomes equal to (315) when the value of k is substituted. 

79 (Page 214). Indeed, when the electron is at rest, the electric 
force in its immediate neighbourhood is E= - -_, • As it is at right 
angles to the surface, there is a normal stress equal to 

80 (Page 215). When, by some disturbing cause, the radius of 
the sphere is increased, the electric stress acting on its surface is 



80 NOTES. 325 

diminislied, as is seen from {131). As the internal stress is supposed 
to remain constant, it will draw the points of the sphere towards 
the inside, so that the original volume will be i-estored. 

We shall next show that the equilibrium would be unstable 
with respect to changes of shape. Consider a deformation by which 
the sphere is changed to an elongated ellipsoid of revolution, the 
magnitude of each element of surface remaining as it was, and each 
element retaining its charge. Then it can be shown that in the in- 
terior, at each point of the axis, there wiU be an electric force 
directed towards the centre if the charge of the electron is negative. 
Let this force be equal to <; at a point just inside the surface at 
one extremity P of the axis. By a well known theorem the electric 
force just outside the surface at the same extremity will be (i-\- a, 
if we denote by — co the negative surface density of the ellipsoid, 
which by our supposition is equal to the surface density of the ori- 
ginal sphere. A surface element at P will be subjected to two nor- 
mal electric stresses, -J-(g -f (of outward, and -^g^ inward; besides these 
there is the constant internal stress which must be equal to \-(a^, be- 
cause, in the original state, it counterbalanced the electric stress. 

Since both q and co are positive, there is a resultant force qa 
directed towards the outside and tending stiU further to elongate 
the ellipsoid. 

In order to prove what has been said about the internal electric 
force, we may proceed as follows. Choose a point A on the semi- 
axis OP, and consider a cone of infinitely small solid angle ds, 
having this point for its vertex and prolonged through it. Let ds^ 
at the point B^, and d6^ at B^ be the elements of the ellipsoidal 
surface determined by the intersection with the cone, ■Q-j and ^^ the 
angles between the line P^ B^ and the tangent planes at the ex- 
tremities, and let Pj be the point nearest A, so that the angle B^AP 
is sharp. Then, since 

the attractions exerted by the two elements on a unit of positive elec- 
tricity at A win be equal to 

cods ^ ads 

and 



It may be shown by geometrical considerations that 
sin ■O'l > sin d-2 , 
from which it follows that, of the two attractions, the second is 
greatest, so that there is a residual force in the direction AB^. A si- 
milar result is found for any other direction of the cone; the total 
resultant electric force must therefore be directed towards the centre. 



326 NOTES. 81-83 

81 (Page 220). The expressions {129) of Note 78 show that, 
if I is different from 1, the value (130) found for the energy must be 
multiplied by I. According to the hypothesis of Bucherer and 

Langevin, l=h ^, which leads to the result mentioned in the text. 

82 (Page 222). If in the equations (200), in which we may 
now omit the terms depending on the resistance and on the external 
magnetic field, we substitute P = D — E, they take the form of a 
linear relation between the vectors D and E, containing their diffe- 
rential coefficients with respect to the time. 

83 (Page 224). Let the effective coordinates of P and Q be 0, 0, 
and x',y',Z] then, by (286), the relative coordinates are 0, 0, and 

-^,y',s'. Hence, if ^,ti,t^ are the values of t at the instants when 

the signal is started from P, received by Q and again perceived at 
P, we have by (284) for the absolute coordinates of the points 
where the signal is found at these moments, 

0,0,0; ~ + wt^,y',z'; wt^,0,0, 

and since the distance from the first to the second is travelled over 
in an interval t^, and that from the second to the third in an inte:-- 
val 4 - t^, 



l + w{t,-t.;)\+ y' + /s = c\t, - Q\ 

By means of these equations t^ and t^ can be calculated. It is 
simpler, however, to consider the quantities 

h=^t,-^x (132) 

and 

V = |4- (133) 

Indeed, the formulae may be transformed to 

giving 

t^ = ~ V^~+7^i^' (134) 

and 



2 



h' = - Vx'' + y'' + /^ (135) 



84, 85 NOTES. 327 

But it appears from equation (288), for which we may now write 

t' = ^t-^x', (186) 

that the variable t^' defined by (133) is the time measured as local 
time of P that has elapsed between the starting and the return of the 
signal. On the other hand, Yx'^ -{- y'^ + /^ is the length L which 
the observer A ascribes to the distance PQ, and —r- is the value of 

h 

the velocity of light which he deduces from the experiment. Equation 

(135) shows that this value wiH be equal to c. 

84 (Page 226). It is sufficient to observe that, as is seen from 

(136) and (132), a clock showing the local time of Q will mark the 
time ty at the moment when Q is reached by the signal, and that, 

according to (134), this time i^' is precisely — ■ 

85 (Page 226). According to what has been said in § 189, the 
mass m which the moving observer ascribes to a body will be the 
mass which this body would actually have, if it were at rest. But, 
the masses being changed by the translation in the manner indicated 
by (305), the real mass will be Jc^m if the acceleration has the direction 
of OX, and Ajw if it is at right angles to that axis. Using the in- 
dices (o) and (r) to distinguish observed and real values, we may 
therefore write 

^(r) = ('«^ ^^'> ^)m(o), 

where the factors enclosed in brackets refer to accelerations parallel 
to OX, OY or OZ. 

On the other hand it appears from the formulae (303) that for 
the accelerations 

so that, if the moving observer measures forces F by the products of 
acceleration and mass, we shaU have 

Now, let two particles with equal real charges e be placed at 
the points of the moving system whose effective coordinates are x^,y^,sl, 
x^', y^', S^ and whose effective distance r is therefore given by the 
first equation of § 171. If these particles had the corresponding po- 
sitions in a stationary system, the components of the force acting on 
the second of them would be 



iTtr'^ ' iTtr " ' iTcr 



(138) 



328 NOTES. 86 

Hence, in virtue of (300), the components of the real force in 
the moving system will be 

(x^' — Xi)e^ 1 (y/ y,>^ 1 _ (g/ - 01 ')e' 

and by {137') the components of the observed force wiU again have 
the values {13^. The observer A will therefore conclude from his 
experiments that the particles repel each other with a force 

and he wiU ascribe to each of them a charge e equal to the real one. 
Let us suppose, finally, that a charge e is placed in an electro- 
magnetic field existing in the moving system, at a point which shares 
the translation. Then, on account of (293), the components of the 
force really acting on it are 

and we may infer from {137) that the components of the observed 
force have the values 

ed;, ed;, ed;. 

It appears from this that, as has been stated in the text, the 
moving observer wiU. be led to the vector d' if he examines the force 
acting on a charged particle. 

86 (Page 230). Einstein uses the substitutions (287), (288) 
and (291) (with I = 1), but he defines the vector u' and the quantity 
Q by formulae slightly difi'erent from (289) and (290), namely by 

u; = ^^^— , u; = — ^-^ — , u; = — ^ — {139) 
1-^V"-^ i-^^V"- ^~*"^"- 

and 

Hence, if the quantities introduced by him and by myself are 
distinguished by the indices E and L, 

"^ ^ ; — ^~ ' p^ = (i - ^'' ^^' "^) 9^' ' 



1 - ^v "^ 



givmg 
and 



Qii\iE= qLuL 



(i - f u;^) p,: = (i - k^ -J- u,) qL = pi. 



It appears from this that the third of the equations (292) remains 
as we found it, but that on the right hand side of the first we get 



78 NOTES 329 

simply q', so that after all the fundamental equations take exactly 
the same form which they have for a stationary system. 

It must be added that, according to Einstein's views, the quanti- 
ties x', y, 3, t' play in all respects the part of the coordinates and the 
time, that, in fact, they must be considered as the true coordinates 
and the triut time in the moving system (if it is examined by an 

observer who moves with it). Thus, the quantities ^, %, -f! deter- 

' ' ^ at ' at ' dt 

mined in Note 75 must actually be regarded as the components of 
the velocity, and the vector u' whose components (139) are equal to 
those quantities, as the velocity itself Again, events taking place at 
different points of space are considered as "simultaneous", if they 
occur at moments characterized by one and the same value of t'. The 
quantity p' is now called "density of charge", and the amount of charge 
existing "at a definite moment t'" in a space whose boundary is given 
by an equation between x', y , z , is given by the integral 



/^ 



Lastly, by means of the hypothesis that the action between 
neighbouring particles is determined by their "simultaneous" positions, 
i. e. by the positions corresponding to a definite value of t' , one gets 
rid of some complications which we encountered in §§ 174 and 175. 

87 (Page 230). Recent experiments by Bucherer^) on the electric 
and magnetic deflexion of ^-rays, made after a method that permits 
a greater accuracy than could be reached by Kaufmann (§ 179) have 
confirmed the formula (313) for the transverse electromagnetic mass, 
so that, in aU probability, the only objection that could be raised 
against the hypothesis of the deformable electron and the principle 
of relativity has now been removed. In the mean time, this principle 
has already been the subject of several important theoretical investi- 
gations. ^) 



1) A. H. Bncherer, Messungen an Beoquerelstralilen. Die experimentelle 
Bestatigung der Lorentz-Einstein'schen Theorie, Phys. Zeitschr. 9 (1908), 
p. 755; Ber. d. deutschen Phys. Ges. 6 (1908), p. 688. 

2) M. Planck, Zur Dynamik bewegter Systeme, Sitzungsber. Berlin 1907, 
p. 542; Ann. Phys. 26 (1908), p. 1. H. Minkowski, Die Grundgleichungen fiir 
die elektromagnetischen Vorgiinge in bewegten Korpem, G-ott. Nachr. Math.- 
phys. Kl. 1908. p. 53. 



INDEX. 

(The numbers refer to pages.) 



Aberration of light, theory of 
Stokes 169—174; theory of Fresnel 
174—180. 

Abraham, M. 32, 36, 213, 214. 

Absorption of light and heat 9 
coefficient of — 68 ; index of — 153, 156 

— bands 155, 156; intensity 155, 156 
explained by impacts 141. 

Alembert, J. d' 17. 
Atom, model of 117; mass 46, 279; 
atom of electricity 44, 46, 280. 

Balmer, J. J. 103. 
Becquerel, J. 311, 312. 
Bestelmeyer, A. 44. 
Black body 69. 
Bockwinkel, H. B. A. 317. 
Boltzmann, L., law of radiation 

72—74, 90. 
Boussinesq, J. 143. 
Brace, D. B. 196, 209, 210, 219. 
Bruhl, J. W. 150. 
Bucherer, A. H. 219, 329. 

Clausius, R. 145. 

Combination of periodical phe- 
nomena 128, 129. 

Conductivity for electricity 7, 10, 
63, 64; for heat 10, 64. 

Corbino, 0. M. 164. 

Cornu, A. 108. 

Corresponding states in moving and 
stationary system, for low velocity 189 ; 
for higher velocity 205, 206. 

Current, electric, 5, 7; displacement 

— 5, 7, 9, 135; — of conduction 7, 8; 
convection — 12, 13; induction — 15. 

Damping of vibrations 259. 
Dielectric displacement 5, 135. 
Diesselhorst, H. 65. 
Dimensions of bodies changed by a 

translation 195, 201, 202, 205. 
Dispersion of light 143, 144, 150, 

151, 152 ; anomalous 156. 
Dissymmetry of Zeeman - effect 

162, 312. 
Doppler's principle 60, 62, 176. 
Drude, P. 10, 63-65. 
Dufour, A. 312. 

Effective charge in moving system 
201. 



Effective coordinates in moving 
system 200. 

Einstein, A. 223, 226, 229. 

Electric force 2, 5, 7, 135; expressed 
in terms of potentials, for a stationary 
system 19; for a slowly moving system 
59; for greater velocity 198. 

Electromagnetic theory of matter 
45, 46. 

Electrons 8; in dielectrics 8, 9; in 
metals 8, 10, 63—67; their charge 
11, 16, 46; mass 38, 39, 44—46; size 
46; heat motion 10, 63; deformed 
by translation to flattened ellipsoids 
210 — 212; deformed without change 
of volume 220; stability of state 214, 
215; model of electron 213, 214. 

Electrostatic system at rest 20; 
moving with small velocity 35, 36; 
with greater velocity 199, 200. 

Emission of light and heat 9. 

Emissivity 69; ratio between . — and 
absorption 69, 89, 90. 

Energy, electric 23; magnetic 23; of 
moving rigid electron 36; of moving 
deformed electron 213, 220; of radia- 
tion 71, 95, 96: intrinsic — of an elec- 
tron 213; kinetic — of a molecule 279; 
equation of — 22; flow of— 23, 25, 26. 

Equations of the electromagnetic 
field for the ether 5; the same, refer- 
red to moving axes 222, 228; for 
ponderable bodies 7; in the theory of 
electrons, referred to fixed axes 12; 
to slowly moving axes 58; to axes 
moving with greater velocity 197; 
for a system of molecules 135, 136, 139, 
140; for a moving system 221 — 223; 

— for electric and magnetic force 17; 

— of propagation 18. 
Equilibrium of radiation70— 72,92. 
Equipartition of energy 90 — 92, 97. 
Equivalent degrees of freedom 112. 
Ether 230; pervades all matter 11, 

174; its immobility 11, 30, 31, 174; 
supposed motion 30, 31; considered 
as having an irrotational motion 171, 
173; as condensed around the earth 173. 

Faraday, M. 1, 98. 

Field of an electron at rest 20; of a 



INDEX. 



331 



moving electron 21, 22, 36, 50, 51; of 
a slowly moving electrostatic system 
35, 36; of a system moving more 
rapidly 199; of a vibrating electron 
54; of a particle with variable electric 
moment 55, 56; of a vibrating par- 
ticle moving with small velocity 59, 
60; with greater velocity 207. 

Fitz Gerald, a. F. 195. 

Fizeau, H. 190. 

Force acting on electric charge 14, 15, 
198; resultant force acting on a system 
of electrons 26, 27, 33; electric forces 
changed by a translation 199; similar 
change of molecular forces 202. 

Frequency, relative 176. 

Fresnel, A. 174; his coefficient 176, 
deduced in the supposition of conti- 
nuously distributed charges 182 — 186 ; 
from the theory of electrons 189, 190; 
for a crystal 192. See: Aberration 
of light. 

Geest, J. 165, 166. 

Gehrcke, E. 259. 

Goldstein, E. 41. 

Gradient of a scalar quantity 4. 

Hagen, E. 80, 81. 
Hallo, J. J. 164, 166. 
Heaviside, 0. 2. 
Helmholtz, H. 30, 136, 143. 
Hertz, H. 2, 57. 
Hull, G. F. 34. 
Humphreys, W. J. 167. 
Huygens's principle 168. 

Image, reflected, of an electromagnetic 
system 130. 

Interference of light in a moving 
system 181, 191; — experiment for 
detecting an influence of the second 
order of the earth's motion 192 — 196, 

Ions 44. [201, 202. 

Irrotational distribution of a 
vector 4. 

Jaeger, W. 65. 

Jeans, J. H. 90; theory of radiation 

93—97, 287. 
Julius, V. A. 128. 

Kaufmann, W. 42, 43, 212. 

Kayser, H. 103. 

Kelvin, Lord 137. 

Kerr, J. 98. 

Ketteler, B. 143. [tion 69. 

Kirchhoff, G. 72, 233; law of radia- 



Langevin, P. 86, 219. 

Laplace, P. S. de 144. 

Lebedew, P. 34. 

Local time 57, 58, 187, 198, 226. 

Lorentz, H. A. 2, 10, 80, 145, 318. 

Lorenz, L. 145. 

Lummer, 0. 75, 259. 

Macaluso, D. 164. 

Magnetic force 2, 5, 7; derived from 
vector potential, for system at rest 
19; for slowly moving system 59; for 
greater velocity 198. 

Magri, L. 146. 

Mass, electromagnetic, of an electron 
38, 39, 43 ; of deformable electron 212, 
213; of electron flattened without 
change of volume 220; of a system 
of electrons 47, 48; ratio between 
charge and — 41, 42, 44, 45, 102, 103, 
212; — changed by translation 205. 

Mathematical notation 3 — 5. 

Maxwell, J. Clerk 1, 2, 27, 30, 67, 
80, 90, 143, 266. [133, 134. 

Mean values in a system of molecules 

Metals; electric conductivity 63, 64; 
conductivity for heat 64 ; ratio between 
the conductivities 65—67; absorption 
by a thin metal plate 81, 82; emission 
by a plate 82—89. 

Michelson, A. A. 190, 191, 192, 195. 

Miller, D. C, 195. 

Minkowski, H. 329. 

Mogendorff, B. E. 107. 

Mohler, J. F. 167. [203—205. 

Molecular motion in moving systems 

Molecules, number of, 167. 

Momentum, electromagnetic, 32; of 
moving electron 36; of deformable 
electron 211. 

Morley, E. W. 190—192, 195- 

Mossotti, 0. F. 145. 

Moving observer measuring lengths 
and times 224 ; studying electromagne- 
tic phenomena in moving system 226; 
in stationary system 227, 228. 

Moving system compared with a sta- 
tionary one 36; for low velocities 189; 
for greater velocities 199, 203—206, 
208. 

Nichols, B. F. 34. 

Onnes, H. Kamerlingh 311. 

Pannekoek, A, 292. 
Paschen, F. 107, 108. 



332 



INDEX. 



Planck, M. 173, 174, 329; theory of 

radiation 78—80, 279, 287. 
Poincare, H. 213, 214. 
Polarization of light obserred in the 

Zeeman- effect 101, 102, 131. 
Polarized light 6, 24. 
Potentials 19, 20, 58, 198. 
Poynting, J. H. 23. 
Pressure of radiation 28, 29, 33, 34. 
Pringsheim, E. 75. 
Products, scalar and vector, of two 

vectors 3, 4. 
Propagation of electric disturbances 
6, 20 ; of light 9 ; along magnetic lines 
of force 157, 158; at right angles to 
the lines of force 159 — 162; in a 
system moving with small velocity 
183—190; in flowing water 190, 191; 
in a system moving with greater 
velocity 205 — 210. 
Purvis, J. E. 108. 

Radiation from an electron 50—52; 
from a vibrating electron 52 — 54; 
from a polarized particle 55, 56; 
from an atom 120. 
Ray of light 169; relative ray 169, 177; 
relation between ray and wave-front 
178; minimum property 179; course 
of relative ray in a moving medium 
178—180; in a moving crystal 192 
canal rays 41; cathode rays 41 
Rontgen rays 52, 152; a-rays 41 
P-rays 41, 42, 212, 329. 
Rayleigh, Lord 196, 209, 219. 
Reflexion by a, moving mirror 

60—62. 
Refraction 9; index of refraction 143, 
153, 156; related to density 144—147; 
of a mixture 147 ; of a chemical com- 
pound 147 — 150; refraction equivalent 
148; double — in a magnetic field 
165; double — that might be produ- 
ced by the earth's motion 196, 219. 
Reinganum, M. 66. 
Relativity, principle of 229, 230. 
Resistance to the motion of an elec- 
tron 49; a cause of absorption 136, 
Riecke, E. 10, 63. [137, 166. 

Ritz, W. 107. 
Robb, A. A. 114. 

Rotation of a vector 4; rotation around 
magnetic lines of force 123 ; of a par- 
ticle in a magnetic field 124 — 126; 



magnetic rotation of the plane of 

polarization 163, 164. 
Rowland, H. A. 13. 
Rubens, H. 80, 81. 
Runge, C. 103, 107, 108. 
Rydberg, J. R. 103; formulae for series 

of spectral lines 104—106, 128, 129. 

Sellmeyer, W. 143. 

Series of spectral lines 103—106. 

Simon, S. 44. 

Solenoidal distribution of a vector 4. 

Statistical method 266. 
Stefan, J. 72. 

Stokes, G. Gr. 52, 169. See: Aber- 
ration of light. 
Stresses in the ether 27, 30, 31. 

Terrestrial sources of light 176, 

181, 209. 
Thomson, J. J. 10, 11, 39, 44, 52, 63, 

117, 120, 288. 

Units 2, 3. 

Velocity of light in ether 6; in a 
system of molecules 142; measured 
in a moving system 224, 225 ; velocity 
of ray in a moving medium 177, 178. 

Vibrations of electrons 9, 49; in a 
magnetic field, of an electron 99, 100; 
of a charged system 109 — 114; of an 
isotropic system 114; of charged 
spheres 115; of a system of four elec- 
trons 120—123; of a rotating particle 
126; electromagnetic vibrations in a 
rectangular parallelepiped 93 — 95. 

Voigt, W. 132, 162, 164, 165, 198, 311. 

Waves, elementary 168, 169, 177; pro- 
pagation of a wave-front 168, 169, 
177; relation between wave and 
ray 178. 

Weber, W. 124. 

Wiechert, B. 52, 165. 

Wien, W. law of radiation 74 — 78. 

Wood, R. W. 167, 312. 

Zeeman, P. 98, 102, 103, 132, 162, 164. 

Zeeman- effect, elementary theory 
98 — 101; more complicated forms 103 ; 
in series of spectral lines 107, 108 ; 
in the radiation from a rotating 
particle 126, 127; inverse effect 132, 
158, 162; dissymmetry 162. 



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I. Band. Elementare Algebra und Analysis. Bearbeitet von H. Weber. 2. Auflage. 
Mit 38 Textfiguren. (XVIII u. 639 S.] 1906. n. Ji 9.60. 
II. — Elemente der Geometrie. Bearbeitet von H. Weber, J. Wellstein und 

W. Jacobathal. 2. Auflage. Mit 251 Textflguren. [XII u. 696 S.] 1907. n.Jil2.— 
III. — Angewandte Elementar-Mathematik. Bearbeitet von H. Weber, J. Wellstein 
und K. H. Weber (Rostock). Mit 358 Textflguren. [XIII u. 666 S.] 1907. n. Ji 14.— 

Wien, Dr. W., Professor an der Universitat Wiirzburg, iiber Elektronen. Vor- 
trag, gebalten auf der 77. Versammlung Deutsoher Naturforscher und Arzte in 
Meran. [28 S.] gr. 8. 1905. Geh. n. JC 1.— 
Wiillner, weil. Geheimer Regierungsrat Dr. A., Professor an der Kgl. Technischen 
Hochscbule zu Aacben, Lehrbuch der Experimentalphysik. In 4 Banden. 
Mit 1104 in den Text gedruckten Abbildungen und Figuren und 4 lithogr. 
Tafeln. gr. 8. 1896/1907: 

Bei gleiehzeitigem Beziige allcr 4 Biindc erniaUigt sicU der Oesamtprcls 
dcs Werkps gfh. auf ii. .11. 44. — , in Halfofrniiz geb. anf n. 51. 50. — 
I'anzcln : 
I. Band. AUgem eine Phyaik und Akustik. 6. Auflage bearbeitet von A. Wulluer und 
A. Hagonbaoh. [XIV u. 1058 S.] 1907. Geh. n. ./« 16.— , inHalbfranzbd. n. ,/<<: 18.— 

II. — Die Lehre von der Warme. 6. Auflage. [XI u. 936 S.] 1896. Geh. u.JilS. 

in Halbfranzband n. Ji. 14. — ' ' 

III. — Die Lehre vom Maguetiamua und von der Elektrizitat mil einer Ein- 

leitung: Grundzuge der Lehre vom Potential. 5. Auflage. [XV u. 1415 S.l 
1897. Geh. n. Ji 18.—, in Halbfranzband n. Ji 20.— 

IV. — Die Lehre von der Strahlung. ."i. Auflage. [XII u. 1042 S.] 1899. Geh. 

n. JI. II. — , in Halbfrauzband n. Ji. 16, — 



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